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Full text of "Study of Stark broadening of high-Z hydrogenic ion lines in dense hot plasmas"

A STUDY OF STARK BROADENING OF HIGH-Z HYDROGENIC ION 
LINES IN DENSE HOT PLASMAS 



By 
RICHARD JOSEPH TIGHE 



A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF 
THE UNIVERSITY OF FLORIDA 
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE 
DEGREE OF DOCTOR OF PHILOSOPHY 



UNIVERSITY OF FLORIDA 
1977 



p^ 



o 



ACKNOWLEDGMENTS 

I would like to gratefully acknowledge the support of an NSF 
Traineeship during the first three years of my graduate study. 

I would like to thank Professor C. F. Hooper, Jr., for suggesting 
this problem and for his guidance and encouragement during the course 
of this work. Also I would like to thank Drs. J. W. Dufty, T. W. 
Hussey, and F. E. Riewe for many helpful discussions. A special 
thanks is due Dr. Robert L. Coldwell for providing guidance in the 
numerical work as well as for lending me several excellent computer 
codes. 

I would like to thank Mrs. Yvonne Dixon for typing the final 
manuscript, and Mr. Woody Richardson for preparing the figures. 

Finally, I would like to thank my wife Janette and my parents 
for the special understanding they have shown during the long years 
of this work. 



11 



TABLE OF CONTENTS 

Page 

ACKNOWLEDGMENTS ±± 

ABSTRACT ' v 

SECTION I. LINE SHAPE FORMALISM 1 

Introduction 1 

Causes of Line Broadening 2 

The Stark Effect for Hydrogenic Ions 5 

A Model for the Plasma 7 

The Line Shape Expression 9 

Time Scales and the Line Broadening Problem n 

Factorization of the Initial Density Operator 13 

The Line Shape in the Quasi-static Ion Approximation 16 

The Liouville Representation 19 

A Perturbation Expansion for R r (w) 20 

A Computational Form for H(w) 23 

The Line Shape Formula 24 

SECTION II. ELECTRIC MICROFIELD PROBABILITY DISTRIBUTION 

FUNCTION 28 

Introduction 28 

The Formal Calculation of T(£) 30 

Introduction of Collective Coordinates 33 

The Collective Coordinate Calculation 36 

Asymptotic Microf ield Distribution Function 51 

SECTION III. DISCUSSION OF THE RESULTS 54 

Introduction 54 

Electric Microf ield Distribution Functions 55 

Stark Broadened Line Profiles 85 

Validity Criteria for this Theory 116 

SECTION IV . CONCLUDING REMARKS 119 

APPENDICES 121 

A. THE INTERACTION V 122 

er 



111 



page 

B. THE ALGEBRA OF TETRADIC OPERATORS 12 4 

C. QUANTUM MECHANICAL PERTURB ER AVERAGES 130 

D. THE PARABOLIC REPRESENTATION 140 

E. CALCULATION OF RADIATOR DIPOLE MATRIX ELEMENTS i 42 

F. THE MANY-PARTICLE FUNCTION r(Au)) 146 

G. A COMPUTATIONAL FORM FOR THE ATOMIC FACTOR 150 

H. NUMERICAL PROCEDURES 15 4 

I. TABLES OF ELECTRIC MICROFIELD DISTRIBUTION FUNCTIONS ... 167 

J. TABLES OF STARK BROADENED LINE PROFILES 220 

LIST OF REFERENCES , 247 

BIOGRAPHICAL SKETCH 250 



IV 



Abstract of Dissertation Presented to the Graduate Council 
of the University of Florida in Partial Fulfillment of the Requirements 
for the Degree of Doctor of Philosophy 



A STUDY OF STARK BROADENING .OF HIGH-Z HYDROGENIC ION LINES 

IN DENSE HOT PLASMAS 

By 

Richard Joseph Tighe 

June 1977 

Chairman: Dr. C. F. Hooper, Jr. 
Major Department: Physics 

Stark broadened x-ray line profiles from highly ionized hydrogenic 
ions, radiating while immersed in a hot dense plasma, are studied. The 
broadening effects produced by the ions present in the plasma are 
treated through the use of static electric microfield distribution 
functions. The microfield distribution functions employed have the 
following properties: (i) the radiating hydrogenic ion may have any 
net charge; (ii) the perturbing ions and electrons in the plasma may 
have different kinetic temperatures; (iii) two species of ions, of 
charge z and z„, may be present in any given ratio. 

Electron broadening effects are treated by the Second-Order 
Relaxation Theory as developed by O'Brien and Hooper. Effects upon the 
electron perturbers due to the fact that the radiator is highly charged 
are included through the use of Coulomb wave functions for the 
perturbing electrons. The present work makes extensions needed for the 
calculation of line profiles from higher Lyman series members. A cutoff 
procedure is employed to simulate the effects due to correlations among 
the electron perturbers. 



v 



Results are presented in graphical and tabular form for both the 
electric microfield distribution functions and for the broadened x-ray 
line profiles. The figures showing the microfield distribution 
functions demonstrate the following effects: (i) when the value of 
the radiator charge is increased, the microfield peaks are raised 
and shifted to lower field values; (ii) an increase of the parameter 

T , the ratio of electron kinetic temperature to ion kinetic temperature, 

R 

causes a similar shift to lower field values; (iii) at low a. values, 
the microfields show a sensitivity to small concentrations of high-z 
ion perturbers. 

The behavior noted for the microfield distribution functions carries 
over to produce similar effects in the line profiles. Although Doppler 
broadening generally dominates the center of the line profiles, the 
wings of the profile demonstrate clearly the behavior noted above for 
the microfield distribution functions. In addition, the structure near 
the line center (i.e. the shoulder in Lyman-a and the dip in Lyman-3) 
shows sufficient sensitivity to the above effects so that it might be 
valuable as a plasma diagnostic tool. 

Directions for future research are discussed. 



vx 



SECTION I 



LINE SHAPE FORMALISM 



Introduction 



Studies of plasma-broadened spectral lines have been quite success- 
ful in determining temperatures and densities for laboratory and astro- 
nomical plasmas. Current investigations involving imploding 
plasmas indicate that the temperature and charged-particle densities for 
these plasmas will be much higher than for those discussed in the above 
references. ' The purpose of the present work is to extend line- 
broadening theories developed previously so that they may be used to 
develop an effective temperature-density diagnostic for the parameter 
ranges expected in plasmas produced in the early laser-implosion 
experiments. 

We consider the broadening of x-ray spectral line profiles emitted 
by highly-ionized hydrogenic ions immersed in hot dense plasmas. In 

Section I, we outline the development of a line-shape formalism which extends 

9 
the work of O'Brien and Hooper to the calculation of higher Lyman series 

members. Again following O'Brien and Hooper, we present in Section II the 

calculation of electric microfield distribution functions which allow 

for different ion and electron kinetic temperatures as well as for 

multiply-charged ion perturbers in varying density ratios. Calculated 

results for microfield distribution functions and for broadened line 



profiles are presented graphically in Section III; the various approxi- 
mations are also discussed. Section IV contains a summary, together 
with concluding remarks. 

Causes of Line Broadening 

The most important mechanisms which cause broadening of spectral 
line profiles are: (1) Natural broadening; (2) Doppler broadening; 
and (3) Pressure broadening. 

(1) In an atomic radiative process, the interaction between the 
excited atom and its own radiation field gives rise to an uncertainty 

in the atomic levels. This uncertainty in the excited-state energy levels 

means that isolated atoms will emit lines having a finite width (referred 

to as the natural width). For hydrogenic ions of nuclear charge Z, the 

natural width is of order, 

2 2 
r * a (aZ) Z Ryd , (1-1) 

Nat 

2 2 
where a is the fine structure constant. Since (aZ) Z Ryd gives the 

order of magnitude of the fine structure splitting for a hydrogenic ion, 

this equation indicates that r,„ is a factor a smaller than the fine 
H Nat 

structure splitting. For the cases considered here, pressure broadening 
effects are at least of the order of the fine structure splitting. This 
means that natural broadening is at least a factor of a smaller than 
pressure broadening in our cases. Consequently, we may safely neglect 
natural broadening in the present work. 

(2) The thermal- motion of the radiator produces another type of 
broadening called Doppler broadening. This effect is due to the Doppler 
shift of the radiation emitted by a moving source. If the velocity 
distribution of the radiator is assumed to be Maxwellian, the resulting 



Doppler line profile is given by 

2 



/ \ 1 r (dl-Un) 

Doppler (U) " TT* 6XP { " ^- } ■ (I " 2) 



where, 



ft 2 - 2kT 2 ^t ^ 

p = co , (1-3) 

Mc 
and T is the kinetic temperature of the radiator. Here, the temperature 
T is the ion kinetic temperature since the radiator is assumed to be in 
equilibrium with the perturbing ions. The mass of the radiator ' is M; 
to is the frequency of the unperturbed transition. The Doppler half-width 
is of order 

2kT. ** 



v >\i i i. 1 

'Tlnnnlfr n i 

Mc 



Doppler !■ _ 2 % (1-4) 



* 10 3 6.^ Z 3/2 Ryd , (1-5) 

where 6. is kT . in keV. From this estimate, it is easily possible to 
obtain Doppler widths of the order of electron volts. In some cases, 
Doppler broadening will dominate all other broadening mechanisms. 

It may be shown (see Equation 8.8 of Reference (12)) that, for the 
case where perturber collisions have no influence on the radiator's 
trajectory (that is, the momentum transfer due to the collisions during 
the time of radiation is negligible) the final Doppler-corrected line 
profile is given by 



00 

■f 



I (to) =\ !„,. . (a)') •_ . (oj-co') dco' , T ,. 

Stark Doppler . (1-6) 



This correction is always valid when the condition, perturber mass << 
radiator mass (corresponding to electron perturbers) , is met. The Quasi- 
static Approximation for the ions, introduced below, allows us also to 






neglect the influence of ion collisions on the radiator's trajectory. 
Hence, the range of validity of the above decoupling of the two broaden- 
ing mechanisms is the same as that of the Quasi-static Approximation for 
the ions (which is well met in the present work - see below) . 

(3) Pressure broadening effects are due to interactions between the 
radiator and the particles surrounding it. These interactions remove 
degeneracies and shift the radiator energy levels. The origin of the 
term "pressure broadening" lies in the fact that this type of broadening 
is sensitive to the density (or pressure) of the particles making up the 
surrounding gas. 

For highly ionized gases, Stark broadening is frequently the most 
important pressure broadening mechanism. However, we may encounter 
situations where Doppler and Stark broadening are equally important or 
perhaps where Doppler broadening dominates. 

In Stark broadening, the radiator energy levels are shifted by the 
Stark effect due to the electric fields of the surrounding ions and 
electrons. The radiator-perturber interaction in the dipole approximation 
is given by, 

V T = eR'e. (1-7) 

Int. 1 ' 

where eR is the dipole moment of the radiating ion (see also Appendix A), 

, -> 
and e. ± s the electric field at the site of the radiator due to the i-th 

l 

perturber. Stark broadening in highly-ionized gases is the primary focus 
of this work. 

Another type of pressure broadening that we must mention here is 
Resonance broadening. This type of broadening occurs when the excited 
radiator is perturbed by ground state atoms or ions which are identical 
to the radiator. Also called "self-broadening," this mechanism is primarily 






important when there is no significant ionization present. 

A multipole expansion of the interaction between two neutral 
hydrogen atoms gives as its lowest order nonvanishing contribution the 
(quadrupole) resonance interaction [see Equation (4-98) in Reference 
(1)]. When the radiators are hydrogenic ions, however, the quadrupole 
term is no longer the lowest order nonvanishing contribution. The 
dominant contribution to broadening between ions is produced by the Stark 
effect (the dipole term) given in Equation (1-7) above. In the present 
work we consider the broadening of hydrogenic-ion lines and encounter 
the situation where radiator and perturber may be identical. Since 
Resonance broadening results from terms higher than dipole in the multiple 
expansion of the radiator-perturber interaction, we neglect it in the 
course of this work. In this case, the neglect of higher terms is an 
approximation whose validity improves as the value of the radiator charge 
increases. Correlation effects due to the monopole interaction of two 
highly charged positive ions produce an effective repulsion. This 
repulsion reduces the probability that higher multipole terms will make 
a contribution (i.e. the repulsion makes close collisions less likely). 
In summary then, although the radiator and perturbing ions may be identical, 
we here neglect the Resonance broadening and consider only the Stark 
broadening interactions. 

The Stark Effect for Hydrogenic Ions 

In a hydrogenic ion, the interaction between the nucleus and the 
single bound electron is a pure Coulomb interaction. This fact has 
several important consequences. The most obvious consequence is that the 
Coulomb problem can be solved exactly: we know the eigenf unctions and 






the eigenvalues for the atomic problem. This solution displays the 
well-known Ji-degeneracy (accidental degeneracy) of the discrete eigen- 
value spectrum for the pure Coulomb system. Thus we must use first- 
order degenerate perturbation theory to calculate the energy level 
shifts due to the linear Stark effect in the hydrogenic case. When this 
calculation is carried out, we obtain the following result for the shift 
of the level specified by n,q,m (for a discussion of the parabolic 
representation, see Appendix D) : 

AE - | -|- nqe, (1-8) 

where a is the Bohr radius; Z is the nuclear charge; n,q are parabolic 
quantum numbers, and e is the magnitude of the electric field. 

In the general case of a non-Coulomb interaction, the ^-degeneracy 
is absent. This means that in nonhydrogenic systems there are no level 
shifts in first order due to the linear Stark effect. For these systems 
the perturbation calculation must be carried to higher order with the 
result that for nonhydrogenic systems, the level shifts due to the Stark 
effect are smaller than in the case of hydrogenic systems. Thus hydrogenic 
systems are more sensitive to pressure broadening effects through the 
Stark effect than are nonhydrogenic systems. For this reason, hydrogenic 
radiators are the optimum choice as emitters of broadened profiles for use 
in plasma temperature-density diagnostics. 

At this point, we must pause to introduce a validity criterion for 
the use of the linear Stark effect in the present work. In the above 
discussion we made no mention of the fine structure contribution to the 
level shifts in hydrogenic ions. An assumption made throughout the present 
work is that the fine structure splitting of the energy levels is negligible 
compared to the Stark shifts. When this assumption breaks down, the 






problem of computing the level shifts becomes very complex and requires 

13 
a numerical solution. The numerical calculation yields shifts, which 

are not linear in the electric field (except, of course, in the large 
field limit) . By equating the Holtsmark shift with the fine structure 
splitting of the upper state of the transition, we may determine a 
validity criterion for the use of the linear Stark effect in line broad- 
ening calculations. If 

n > 2.4 x 10 16 (V 5/2 (1 - 1 ) 372 , (1-9) 

e n n 

where n is the electron number density; Z is the nuclear charge of the 

radiating ion; and n is the principal quantum number of the upper state, 

then the linear Stark effect is appropriate for our calculations. 

As we will discuss in a later section, several cases considered in 

this work approach the limit of validity given above. 

A Model for the Plasma 

A plasma is an electrically neutral gas in which the temperature is 
high enough so that some degree of ionization is present. For the purpose 
of this work, a plasma is defined to be a gas which is ionized to the 
extent that Stark broadening is the most important pressure broadening 

mechanism. 

4 
Although normal laboratory plasmas at temperatures -10 K may have a 

significant number of neutral atoms present, the work to be reported 

here is aimed at treating dense, hot plasmas of laser-fusion experiments 

in which case all neutrals will be strongly ionized. Therefore, our 

plasma is composed of electrons and ions of various ionization stages. 

The radiation we choose to study is the Lyman series of lines from the 

hydrogenic ionization stages of various heavy atoms. The specific atoms 



8 

are selected on the basis of the predicted population densities of their 
hydrogenic ionization stages. 

In the formal development which follows , we employ a model in 
which the plasma contains only one radiating hydrogenic ion. This model 
assumes that the individual radiative processes which add together to 
produce the line profile occur independently of one another. For the 

present work, this is a reasonable assumption since radiative processes 

15 
in plasmas will, in general, add incoherently. 

In the above model, it may appear that we are neglecting all the 
contributions from processes where two radiators interact with each 
other. However, the dominant contribution from such processes, namely 
the Stark broadening interaction, is contained implicitly in the treat- 
ment of ion broadening through use of electric microfield probability 
distribution functions (see the discussion of this procedure below) . 
This statement is based on the fact that in the microfield calculation, 
the ion perturbers are treated as classical point particles with no 
attention given to quantum problems such as particle identity or internal 
structure. This means that even though a perturber may be identical to 
the radiator, it produces Stark broadening effects which are independent 
of its internal energy state. 

An additional feature of the model is that we fix the radiator at 
the origin of the coordinate system. This allows us to concentrate on 
Stark broadening and add Doppler broadening at a later step. In summary, 
our model consists of the single hydrogenic radiator fixed at the origin, 
surrounded by a gas of perturbing electrons and ions. When this model is 
employed in an ensemble average to compute a line profile, the result will 
represent a profile emitted by an actual system containing many essentially 
independent radiators. 






The Line Shape Expression 

The total power emitted by a quantum system in a spontaneous 

electric dipole transition is given by 

, 4 

4to_, „ , , , ,2 



a \ I | < b | exj j a> J ; (1-10) 

J 



El 3c 3 



oj . is given by the Einstein formula, 
ab 

W ab = ^."V 7 "' (I_11) 

and c is the speed of light. Since we are considering an emission process 
only, the initial state of the system has energy E and the final state 

3. 

has energy E , where E > E, . The sum over j includes all the components 
of the dipole moment of the system. 

To make the connection between Equation (1-10) and the spectrum 
emitted by a plasma, we must carry out an ensemble average of this equation. 
This means we must average over the initial states a by including a weight 
factor p „ In addition, we must sum over final states b_. Then the power 

3. 

spectrum emitted by a radiating system becomes 

4w , „ 

P (w) = — ^- I 6(u)-oj , ) |<b|exj|a>r p . (1-12) 
j ab a 

3c a,b,j 

The delta function ensures that the transition conserves energy. We now 

define the line shape function I(u) : 

I(io) =1 6(d) - oj ) |<b|exj | a> | 2 p . (1-13) 

a,b,j 

Then P(w) = (4w , /3c ) • I(w) where I(co) has the property of being 
ab 

applicable to absorption as well as emission processes. Now consider the 
Fourier transform of I(co) , 



10 
$(t) = \ dtoe "" l " I (to) , 



- s 



or 



't) - ^ ' : 



$(t) = L . e ab <b exj a> p . (1-14) 

a,b, . ' n ' ' a 

j 

From this we can see that $(t) = [$(-t)]*. Consequently I(to) may be 
expressed by 

oo 

I (to) = -tt" 1 Re \ dt e iut $(t) . (1-15) 

o 

Also $(t) may be written in the form 

l(t) = I e'^ab* 1 <b|3|a>-<a|d|b> p , (1-16) 

a,b 

-> 
where d is the electric dipole moment of the total system. We define p 

9. 

to be an operator that acts only on the initial states a so that we may 
move it inside the matrix element: 

*(t) = I e" 10J ab t <b|d|a>-<a|pd|b> . (1-17) 

a,b 

Inserting the definition of to n we obtain 

ab 

$( t ) = I e~ i(E a" E b )t/ri <b|d|a>-<a|pd|b> . (1-18) 

a,b 

Since the E's are the eigenvalues of the Hamiltonian for the system we 

may also move the exponentials inside the matrix element: 

»(t) - I <b|d|a>.<a|e~ iHt/H "pd e iHt/rr |b> , (1-19) 

a,b 

•> -> 

-J <b|d|a>-<a|x(t)pd T + (t)|b> , (1-20) 

a,b 

where the time-development operator T(t) is defined by 

T(t) = exp {-iHt/Ti} . (1-21) 






11 

The sum over the states a and ib Is just a trace operation: 

«(t) = Tr {d-T(t)pd T (t)} ; (1-22) 

$(t) is thus obtained by performing a trace over the states of the total 

quantum system. We may observe that <f>(t) is the autocorrelation function 
for the electric dipole moment of the system. 

Time Scales and the Line Broadening Problem 

We demonstrated above that 1(a)) can be written in terms of the 
following transform: 

I(w) = -ff" 1 Re \ dt e 1U)t *(t) . (1-23) 



\ 



Since $(t) is an autocorrelation function, we expect it to be a smoothly 

decaying function for large t. This means that for t >x (where x is 

R K 

some critical time for the broadened emission process) , the exponential 

will oscillate so rapidly that the contributions to the transform will 

vanish. This is a statement of a familiar property of Fourier transforms, 

namely 

Aid T n < 2ir . (1-24) 

R 

Most of the contribution to I(w) will come for times t less than the 

critical time x . This suggests that an appropriate characteristic time 
R 

for the broadened emission process will be defined as 

Tr e ^w • »- 25) 

where Ato , is the width of the resulting (experimental or calculated) 
o £ ar k. 

Stark line shape. 

Processes within the system having characteristic times greater than 

t will take place so slowly that they may be regarded as static. 

R 



12 

Approximations of this type are useful so long as their region of 
validity covers most of the line profile. This will nearly always be 
the case for the motion of the ion perturbers. A characteristic time 
for the ions is given by 

T_ = -~ ^ i- , (1-26) 

Ions v to f - v 

av p (Ions) 

where A^ is the Debye length for the plasma: V is the average thermal 
D av 

velocity of the ion perturbers. Essentially, T T gives the duration of 

r " Ions b 

a radiator-ion encounter: lo , T N is the plasma frequency for the ions. 

p (Ions) 

In most cases of interest, the ion plasma frequency tii ,_ , is much 

p (Ions) 

less than the width of the line profile: 

(0 , T K < Aw_„ . T T > T n (1-27) 

p (Ions) Stark; Ions R 

In the formal development, we introduce an approximation in which ion 
motion is neglected and ion broadening effects are treated by averaging 
over static configurations of the ion perturbers. In this approximation 
(known as the Quasi-static Ion Approximation) , the ions provide a static 
electric field which splits out the atomic levels of the hydrogenic 
radiator. The average over the ions is carried out in a final step of 
the line profile calculation by an integration over an electric microfield 
probability distribution function. 

There are, however, ions in the plasma whose velocities are greater 

than V . Characteristic times for these (dynamic) ions are less than T„ 

av J R 

and the effects of their motion may not be negligible. Equations (1-24) 
and (1-26) allow us to make an estimate of the limits of validity for the 
Quasi-static Ion Approximation: 

Aw t t < 2tt 

Ions — ' 

or, 



13 

Aw.< to ,, N . (1-28) 

— p (Ions; 

By this we mean that for frequency separations Aw (measured from the 
unperturbed transition frequency) less than oj , broadening effects due 
to dynamic ions will become important. For all the cases considered in 
this study, the ion plasma frequency is so small compared to the width of 
the profile that ion dynamics are important only over a small frequency 
range at the very center of the profile. Therefore, ion dynamic effects 
are neglected throughout the course of this work. 

When we apply the same analysis to the perturbing electrons, we 
observe some striking differences. First, the electron plasma frequency 
is -42 times greater than the plasma frequency for protons. In the case 
we study here, the electron plasma frequency falls out in the wings of 
the line profiles. Electron dynamics then are important over most of the 
line profile and static approximations are of interest only in the far 
wings. This means that the time dependence of the electric field due to 
the perturbing electrons requires quite a different treatment from the 
one employed when dealing with the ions. In the present work, electron 
broadening is treated by a second-order time-dependent perturbation 
calculation. 

Thus a hierarchy of time scales is present for the line broadening 

problem: 

T < T < x (I _ 29) 

Electrons R Ions 

This hierarchy gives justification for the two differing theoretical 

approaches to the broadening produced by the electron and ion perturbers . 

Factorization of the Initial Density Operator 

The initial density operator p is chosen to be the canonical 






14 
Boltzmann operator given by 

p = e~ BH / Tr {e _6H } , (1-30) 

where g = (k T)~ and H is the Hamiltonian operator for the total system. 
B 

H = H° + H° + H° + V + V. + V . . (1-31) 

r l e er lr ei 

Here, H is the unperturbed Hamiltonian for the radiator, H. is the kinetic 
energy of the' ions plus their (ion-ion) interaction energy, and H g is 
similarly defined for the electrons. The V's are the respective inter- 
action energies. 

We now introduce an approximation present in most line broadening 
calculations. We assume that the operator p has only diagonal matrix 
elements between the initial states of the transition. Furthermore, we 
assume that p may be factored in the following manner: 



P = P P-P 
r i e 



(1-32) 



In order to define the factors on the right side of Equation (1-32) 
we present the following argument. We first regroup the terms in Equation 
(1-31). After performing this regrouping, we may write H in the following 
form 



. „0 



H = H + H. + H +V , (1-33) 

r i e er 



where 



H = H° + V a) , (1-34) 



r r lr 
and 

H. = H° + vf 0) + V . . (1-35) 

11 lr ei 

In this expression, V. is the monopole contribution to V. , namely 

V< 0) = X z SL. (1-36) 

lr A p r 
P 



15 

In this equation, y( =z ~l) is the net charge of the radiator and z is 

the charge of the ion perturber. It is important to note that in this 

form this term is independent of radiator coordinates. 

The term V. gives the dipole contribution to V. and is combined 
ir ir 

with H since it contains the radiator position operator: 

V a) = eR-e. . (1-37) 

ir 1 

Terms in the multipole expansion of V. that are higher than dipole are 

not considered since the present work will employ the dipole approximation 

to the radiator-perturber interactions. 

If we now neglect the V term in Equation (1-33) we may obtain the 

approximate factorization of Equation (1-32) . The following definitions 

are possible: 

p = e" 3H r / Tr {e~ 3H r} , (1-38) 

r r ' 



p. = e" 3H l / Tr {e" 6H i} , (1-39) 

where the prime here indicates that we consider the term V . to produce a 
1 ei 

Debye shielding effect in H.. Therefore, we drop V . and now employ only 

shielded interactions in order to leave El independent of electron 

l 

J« «. 17 

coordinates. 

p e = e~ 6H e / Tr {e~ 6H e} . (1-40) 

Neglect of the term V implies that while the electrons produce 
broadening effects, they do not alter the initial distribution of radiator 
states. The factorization of p is a procedure which need not appear in 

the formal development of modern line broadening theories based on kinetic 

1 8 
theory. These theories, however, sometimes make a factorization for 



reasons of computational convenience, 



16 
The Line Shape in the Quasi-static Ion Approximation 

Combining Equations (1-22) , (1-23) , and (1-32) we obtain for the 
line profile 

oo 

I(io) = ^ _1 Re \ dte lwt Tr {d T(t)p p.p dT + (t)} . (1-41) 
j r i e 

o 

The trace is to be evaluated using states of the entire quantum system 

of radiator, electrons, and ions. The most convenient set of states for 

this calculation is a set of product states. Each product state will 

consist of a one-particle state for the radiator and a many-particle state 

for the gas of electrons and ions. We now define this product state: 

|a> = | y> x |a> , (1-42) 

where p represents the one-particle radiator state and a represents a 
many-particle state for the electrons and ions. 

■4- 

Now we introduce the familiar restriction that d is the dipole 

1 ■+ 

moment operator for the radiator only. Furthermore, we restrict d to 

have nonzero matrix elements only between a specific set of upper and 
lower states; hence we will consider only line radiation of a specific 
transition, In our case, we consider only the Lyman series of transitions 
for hydrogenic ion radiators. 

Now the Quasi-static Ion Approximation is introduced as follows. We 
first consider the commutator of HT with H as given in the expression 

ih 4~ Ht - [H:, H] . (1-43) 

dt l l 

— 1 18 
Smith has shown that this commutator is proportional to m. . Since the 

thermal velocity of the ions also scales as m. , it is evident that the 

limit of infinitely massive ions corresponds to the case where the ions 

are static. This same limit also implies that the commutator above vanishes: 



17 
[H:, H] = . (1-44) 

Several results follow immediately from this equation. The first 
consequence of Equation (1-44) is that we may write 

T(t) = T (t) T r (t) ; T + (t) = T^ (t) T i + (t) , (1-45) 



where 



T.(t) = e- itH i /B ; T (t) = e - itH er /E , (1-46) 



i er 



and 



H = H + H° + V (1-47) 

er r e er 

Another consequence of Equation (1-44) is that p. commutes with p r and 

T (t). Using these results we may simplify Equation (1-41), with this 
er 

result for the line profile: 

CO 

I(u>) = u" 1 Re \ dte ±a)t Tr {p.d T (t)p D d T* (t)} . (1-48) 
J l er r e er 

o 

Note that T. operators have been commuted and canceled. Next, we insert 
x 

■> ■+ 
a delta function 5(e - e.) into the trace of Equation (1-48), along with 

■> 
an integration over the variable e. This step is valid so long as the 

delta function (which contains the ion coordinates) commutes with all the 

other operators inside the trace. The vanishing of the commutator in 

Equation (1-44) ensures the validity of this step. After this insertion, 

we are free to reverse the order of the trace and integration operations: 



oo co 

\ de 7r~ Re \ 
o o 



■+ -y 

I(w) » \ de TT - "'" Re \ dte 1W Tr {p.6(e - e.) 



.d T (t)p p d T + (t)}. (1-49) 

er r e er 

The effect of inserting the delta function here is that we may replace 

e as it appears in T (t) and p by the integration variable e. After 
i er r 

making this replacement we are free to perform a partial trace over the 






18 
ion coordinates in Equation (1-49). The line profile in the Quasi-static 



Ion Approximation is then given by 

I(u) - \ de Q(e) J(o),e) , (1-50) 

where 



Q(e) - Tr. {p.6(e - e.) } (1-51) 

defines the electric microfield probability distribution function for the 

static ions: Q(e) gives the probability of finding an electric field £ at 

-> 

the site of the radiator. Now J(w,e) is defined in terms of a trace over 
radiator-electron product states. 

oo 

J(u,e) = tt" 1 Re \ dt e lt0t Tr {d-T (t)p p d T + (t)} . (1-52) 
J er er re er 



-*- 

The operators T (t) and p are now functions of H (e) where 
r er r r 

->- He -»■ 

H (e) = H° + eR'e , (1-53) 

r r 

-*■ ■+ 

and £ is the electric field due to the static ions. Here J(w,£) gives 

■4- 

the electron broadened profile emitted by an ion in an external field £. 
Ion broadening effects are included when the ion microfield integration 
of Equation (1-50) is performed. 

Before continuing, let us summarize the approximations we have made 
thus far: 

i) The dipole approximation to the radiator-perturber interactions 

[Equation (1-7)]; 
ii) The factorization of the density operator [Equation (1-32)]; 

iii) The Quasi-static Ion Approximation [Equation (1-50)]. 

These approximations have been standard line broadening approximations, 
the validity of which will not be tested in this work. It should be 
pointed out, however, that the first two restrictions may be, in principle, 



19 

removed. (See References 4, 17, 19, and 20.) Our previous discussion 
of ion dynamics and the validity range of the Quasi-static Ion Approxi- 
mation [see Equation (1-28) and the subsequent paragraph] indicated that 
for the cases considered here, the Quasi-static Ion Approximation may be 
used with considerable confidence. 

The Liouville Representation 
In this section we introduce a notation which formally simplifies 
the calculation of J(w,e). Consider an arbitrary operator f whose time 
dependence is generated in the following manner: 

f(t) = T(t)f(0)T + (t) - e " iHt/H f(0) e 1Ht/H . (1-54) 

If we take the time derivative of f(t), we obtain, 

ih^ f(t) = [H,f(t)] = Lf(t) . (1-55) 

This equation defines the Liouville operator L and its operation on an 
arbitrary operator. We may solve this equation formally for f(t): 

f( t ) = e " iLt/R f(0) • (1-56) 

The Liouville representation as introduced in Equations (1-55) and (1-56) 

(see also Appendix B) is essentially the "doubled atom" representation 

2 
of Baranger. The advantage of this notation is that it gives a "short- 
hand" with which to carry out formal manipulations. That is, we may 
formally perform the integration of Equation (1-52) with the following 
result for J(oj,e): 

J(u,e) = -TT -1 Im Tr {d'K ( w )p p d} , (1-57) 

er er r e 

where K (to) is defined by 
er 



5iu)t -il 
dt e e 



K (co) = -i | dt e iU,L e~ iLt/E = {w-L/h} X . 
er 



20 
The operation of L is defined by 

Lf = [H , f ] , (1-59) 

er 

where H is defined in Equation (1-47) . We now formally carry out a 
er 

partial trace over the states of the perturbing electrons. 

J(u),e) = -Trim Tr {t R r (ai)p r 1} , (1-60) 

R Go) = Tr {K (ai)p } = <K («)> . (1-61) 

r e er e er 

The last equation defines our use of the bracket notation to indicate 
an average over the perturbing electrons. 

The operator R (to) is called the effective-radiator-resolvent. 
Although this operator is a function of radiator coordinates only, it 
includes broadening effects due to the dynamic electron perturbers. 
Equations (1-58) and (1-61) give the formal definition of R (oj) . The 
goal of sections to follow is the development of a useful method for 
calculating R (oj) . 

A Perturbation Expansion for R (lo) 



In this section we describe a method for approximating R (to) , the 
effective radiator resolvent operator. The expression we are interested 
in is given by 

R.(u>) = <K er (ui)> = <{a)-i./H} _1 > , (1-62) 

where L is given by 



L - L Q + \ I . (1-63) 



This corresponds to 



21 

H = E n + XV = H° + H° + XeR-e + XV . (1-64) 

er I r e er 

Here X is a coupling constant introduced for convenience (later we will 

let it equal unity). The procedure we now follow is the same as that 

employed by Duf ty in Reference (21) . The first step in this procedure is 

to make the following definition: 

<K (a) > = {io-L°/lT - XL. /ft - HU)}" 1 . (1-65) 

er r lr 

This expression now formally defines the operator n'(to) : H(co) is a function 

of radiator coordinates only, but contains broadening effects due to the 

perturbing electrons. We now assume that the operator H(co) has an 

expansion in powers of the coupling constant, namely, 

H(u>) = H (0) (u>) + XH (i) (o)) + X 2 H (2) (a)) + ... . (1-66) 
The next step involves expanding both sides of Equation (1-65) in 
increasing powers of X and equating like powers to identify terms in the 
perturbation expansion of tf(w) . The left side of Equation (1-65) may be 
expanded in a Lippmann-Schwinger expansion by employing Equations (1-62) 
and (1-63): 

<K er (o))> = <R Q (w)> + Xft" 1 <R (io)L I R (a))> 

+X 2 K" 2 <R (a,)L I R (co)L I R (o J )> + ... . (1-67) 

Several identities given in Appendix B help to simplify this result. 

<K (to)> = R°(oj) + Xn~ i R°(u))<L T >R (a)) 
er r r I r 

+X 2 E _2 R°(a))< L i R (oj)L i >R°(w) + ... , (1-68) 

where 



22 
R°(a)) - {oj-L^/R}" 1 ; R ((o) = {oj-^/E}" 1 . (1-69) 

The right hand side of Equation (1-65) may be expanded in a Taylor series 
in A with the aid of the following operator identity. For an operator A, 

d -1 -1 dA -1 . „ . 

^A =-A -A . (1-70) 

The expansion of the right hand side of Equation (1-65) is given by 

{03-L°/n - AL. /K - H(w)}' 1 = {w-L°/R - f/ (0) (a)) } _1 
r lr r 



+ A{w-L°/R - H (0) (a,)} _1 [L. r /h + H (1) ( W )]{oD-L°/h - H (0) (w) r 1 
+ A 2 {w-L°/R - H^Ca))} -1 [L. r /h + H (1) (co)]{oo-L°/K - tf (0) (co) T 1 
x[L ir /E + n' (1) ( u ) ] {u>-J, /E - H (0) (u) } _1 

+ A 2 {w-L°/R - H^Wf 1 H (2) (a>){ai-L°/ti - H (0) ( W )} _1 + ... (1-71) 

Now by comparing Equations (1-68 and (1-71) we may identify the terms H 
(oi) appearing in Equation (1-66) : 

H (0) («) - ; (1-72) 

hH (1) (a) = <L er > ; (1-73) 

H 2 H (2) (o)) = <L er R Q (a,) L er > - <L £r > R°(u>)<L er > . (1-74) 

Before proceeding, we state here (and prove in Appendix C) that, as 
a consequence of making the dipole approximation for the radiator-electron 
interation V (see Appendix A) , the indicated average in Equation (1-73) 
vanishes : 



<L > = . (1-75) 

er 



23 

This means that if we retain the lowest order nonvanishing contribution, 
f/(w) is given by 

H(u>) = R~ 2 <L R.(o>) L ■ > , (1-76) 

er er 

with the final result for R (w) : 

R (to) = {o)-L°/h - L. /h-H~ 2 <L R„(ui) L >}~ 1 ; (1-77) 
r r lr er er 

R (its) expressed in this form is referred to as a second-order resolvent, 
r 

A Computational Form for H(co) 

In this section we wish to develop a form for the second-order result 

(2) 

n (to) which is convenient for computation. That is, we wish to develop 

a matrix representation of the following expression, 

2 (2) 
h fT ; (w) f =<L R n (o>) L f> , (1-78) 

er er ' 

where f is an arbitrary radiator operator. We may insert the integral 

definition of R_(w) and obtain 



CO 

dt e 



h 2 H (2) (co) f = -i \ dt e^ <L e- lL t/Tl L f> , (1-79) 

er er 



CO 

-I 



.„ icot , „ -iH„t/n TT » -iH n t/n TT - iH_t/ti TT 

dt e {<V e V f> - <e V feO V> 

er er er er 

b 



-iH_t/h t TT !H rt t/n -iH ft t/h * T7 iH.t/h Tr , 

-<V e f V e > + <e f V e V >} /T onS 

er er er er . (1-80) 

We concentrate on the first term in this expression. Denote this term by 

W(w)f; then, 

00 

n W(oi)f = -l I dt e <V e V f e > . (1-81) 

J er er 

o 

We now take matrix elements of this expression between free radiator 






24 



21 A . . 
eigenstates. Continuing m this manner, we may extract a matrix 

representation of the tetradic operator (see Appendix B) W(ai) . The 

9,22,23 



result of this calculation is' 

H 2 W(co) , , = -i<5 , t. f A * -^■■"■ t <v " n ii-.i^w ' (1-82) 
pv;y v vv i 

M o 

where 



j dt e iAu y»v t <V uy » V u » u '(t)> 



A %"v " W "V'v and >p' Ct) = e_iHet/K Vv elHet/R ■ (I ' 83) 

The calculation of the electron-averaged quantities appearing in Equation 
(1-82) is discussed and carried out in Appendix C. The result for the 
full expression in Equation (1-80) is given by 



E 2 H (2) (w) =6 , E R ,,-R" „ , r(Aoj „ ) 



w „ up y u' y v 



-6 , E f , ,,-R „ r(-Ato „) 
py .i v'v" v v yv" 



+ R ,-5 , {r(-Aaj ,) - r(Aoi , )} . (1-84) 

pp VV yv' y v 

In this equation, R is the dipole moment of the bound radiator electron. 

The complex function T(Aco) contains many-particle effects due to 
the perturbing electrons. Its definition and the details of its calculation 
are given in Appendix F. The calculation of the radiator matrix elements 
appearing here is given in Appendix E. 

The Line Shape Formula 

We now incorporate the results of our calculations into the line 
shape expression. Recall that the line profile is given by 

DO 

I(u)) = l de P(e) J(tt>,£) , (1-85) 





25 
where 

P(e) = 4 ire 2 Q(e) , (1-86) 

■+ ■+ 

and we have made use of the fact that Q(e) is an isotropic function of e. 

J(io,e) = -tt~ im Tr (d-R (co)p d) . (1-87) 

r r r y 

The effective radiative resolvent operator R (a) contains averaged 
electron broadening effects: 

R r (u) = {u)-L°/h - L ±r /n - H(oj)} -1 . (1-88) 

To second order in the radiator-electron interaction, H(to) is given by 
Equation (1-84). 

We now insert free radiator eigenstates in order to evaluate Equation 
(1-87). 

J(oj,e) = -7T Im {Z Z <v|a r L>. [R (ic)J , , , <u' Ip d|v'>. (1-89) 

i i f r pv;y* v 1 i*r ' 

This equation is the final form of Equation (1-12). These two equations 
are still similar in that there is a weighted average over initial states 
of the transition along with a sum over final states. Previously we 
indicated that the weighting factor p is a diagonal operator acting upon 
the initial states of the transition. Also, the matrix elements of J are 
restricted to have non-vanishing results only between initial and final 
states of specific transitions. These facts allow us to write for J(w,e): 

j( Ui e) = -u _1 Im J J f , [<f|d|i> • [\(w)] if;i . f , <i'|d|f> P ., 

+ <i|d|f> • [R»] fi;i , f , < i , |d|f> P .,]} . (1-90) 

At this point, we introduce the No-Quenching Approximation. Mathe- 
matically, this approximation states that the matrix elements of the time 






26 
development operator for the system taken between initial and final states 
of the transition must vanish. That is, 

<f|e _1Ht/n |i> = 0, (1-91) 

where H is the Hamiltonian for the system. This matrix element is 
proportional to the probability amplitude for a process where the inter- 
actions within the system cause a radiationless transition from the upper 
state to the lower state. The No-Quenching Approximation, therefore, is 
invalid when the broadening interactions mix the upper and lower states 
of the transition. This means that broadening effects must be small 
compared to the separation of adjacent radiator energy levels. This 
statement must be considered either to be a weak collision assumption or 
it must restrict us to calculate only isolated line profiles. We simply 
state here that if we expand the factor R (a) f ..., f , [appearing in 
Equation (1-90)] according to previous definitions we would find factors 
like that in Equation (1-91) , but with H replaced by an effective Hamiltonian 
operator containing averaged electron broadening effects. From this 
procedure we see that the No-Quenching Approximation causes the second 
term in Equation (1-90) to vanish. 

We now point out that in this work we consider only Lyman series 
transitions so that we need not sum over lower states. 

-1 



J(co,e) - -n Im E <l|d|i> • [R (w) ] , <i'|d V [l>p i , (1-92) 

ii 1 ' 

where we define [R (co) ] . . , by 

r ii' 

ry<-)] ±1 , - ty^n^! • ' ci-93) 

By defining 



D., . = <i'|d|l> • <l|d|i> , (1-94) 



27 
we obtain a simpler form for J(co,c): 

JCu.e) - -iT Im E D.,. [R Cm)].., p., , (1-95) 

... 1 1 r n l 
n 

where D is a scalar matrix we calculate in Appendix E. The sum over i 

corresponds to ordinary matrix multiplication. The sum over i' represents 

the desired thermal average. 

The above restrictions simplify the definition of R (oj) : 

[R (oj)].., = {Aa)-eeR Z .,/h - HCu))..,}" 1 , (1-96) 

r n n n 

where Aw = w-w. f indicates that we measure frequency in terms of separation 
from the unperturbed frequency; R is the z-component of the position 
operator for the single bound radiator electron. 

H(to).., = h~ 2 r(Aoj) E R..„ • R.„., . (1-97) 

n .„ n l l 

This completes the discussion of the line shape formula. Actual 
details of the numerical procedures involved in computing line profiles 
will be discussed in later sections. In summary, let us list the major 
approximations made in reaching this point: 

i) The dipole approximation to the radiator-perturber interactions; 
ii) The factorization of the density operator; 
iii) The Quasi-static Ion Approximation; 
iv) The second-order perturbation calculation for the effective 
radiator resolvent operator; 
v) The No-Quenching Approximation; 
vi) No lower-state broadening (this is not an approximation for 
the Lyman series) . 



SECTION II 



ELECTRIC MICROFIELD PROBABILITY DISTRIBUTION FUNCTION 



I 



I(w) = I de Q(e) J(oj,e) . (II-l) 



Introduction 

In Section I we indicated the steps involved in arriving at the 
Quasi-static Ion Approximation. As a result of making this approxi- 
mation, the line broadening problem is greatly simplified. That is, 
the line profile may be written 






Included in the function J(oi,e) are the electron broadening effects, 






calculated for the radiator placed in the static electric field of the 
perturbing ions. This calculation is discussed in detail both in Section 
I and in the Appendices . The electron-broadened profile is averaged over 
all possible values of the ion field to produce the final Stark profile. 
This average is performed when we integrate over the electric microfield 
probability distribution function Q(e). The probability Q(e) of finding 

-4- 

an electric field e at the site of the radiator, was defined in Equation 
(1-51). We now write Q(e) in the following form: 

Q(e) = z" 1 I. ..I expl-^VCr^...,^)} 6 ( E- \ ?.)dr" N . (II-2) 

Here z is the conf igurational partition function and we average over con- 
figuration space; V(r , ...,r ) is the total potential energy of the ions, 
including their interactions with the radiator; and e. is the field at the 

-1- 

28 






29 
-*■ 

site of the radiator due to the perturber located at r . : 

->■*•-*• -> 

E. - - V. V(r_ ...,0 . (II-3) 

i x * 1> N 

The usual minus sign is cancelled here because we express the gradient 

->- 

in terms of r . . 
x 

At this point we wish to recall some of the features of the model 
that we employ when performing the microfield calculation. There is a 
hydrogenic ion (radiator) of charge x e fixed at the origin of our coordinate 
system. The plasma perturbers consist of electrons along with two species 
of ions of charges Z and Z„, respectively. Overall charge neutrality of 
the plasma is expressed by the following relation 

n = Zn, + Z_n„ , (II-4) 

e 11 2 2 

where n is the number density of the electron perburbers and n (n„) is the 
e 1 z 

number density of the charge Z (Z ) specie of ion perturbers. In addition 
to providing charge neutrality, the electron perturbers are assumed to 
produce a Debye shielding of the various interactions between the ions in 
the plasma. That is, in the present work, we restrict our consideration 
to the "low-frequency" microfield distribution function. 

We insert the integral definition for the delta function in Equation 
(II-2): 



'H 



Ke) : \...j (2w) _3 exp{-B i V + i£-(e- Ee ± ) }d£dr . (11-5) 



Since £ is an arbitrary vector, Q(e) cannot depend on the direction of £. 

We may then perform the angular integration, arriving at the relation, 

oo 

Q( e ) = (2TT 2 e) _J - \ T(£)sin(e£) £ d£ (II-6) 



'' 



o 
where 



1 \...| exp{-6.\ 



30 



T(4) = Z J \...\ exp{-6.V -BlIA'*J dr N . (II-7) 



1 



Also, since the ions are distributed isotropically in the plasma we may 
write, 

2 
P(e) = 4tte Q(e) 

CO 

= 2ir e \ T(£)sin(e£) £ d£ . (II-8) 



M 



The transform in Equation (II-8) is performed numerically; the 
calculation of T(Jt) forms the principal topic of this section. 

The Formal Calculation of T(£) 



The explicit form for V(r , . . . ,r ) is given by 

N. „2 2 M 2 2 

-*• . -> lZe _ i 2 Ze _ 

V(r-,...,r„) = E -^— e r iJ /X + E — — e X ^ X 
1 N . . r.. „ r 

i<j ij m<n mn 



N, N. _ _ 2 N 2 

1 2 Z Z„e ,, J- X^-, e _„ ,, 

+ j E -LL- e -r jm /^ E -J— e r J0A 

j m jm 3 J u 



XZ 2^ "%oA 



+ E — — e l «' n . (H-9) 

m mO 

We specify a convention where i and j are reserved for the specie of 

density N and charge Z . Subscript refers to the radiator. We now 

introduce the quantities: 

yZ e 
W. n - — i— e~ ar J0/ A ; (11-10) 

^° r j0 

2 
w = ^2l_ e -^ m0 /A 

mO r n ; (H-ll) 

ml) 

N l N 2 
V - V Q + E W + I W mQ . (II-12) 

3 m 






31 

The W's represent a short-range part of the central interactions between 
ions and the radiator. All the noncentral and long-range central inter- 
actions are compressed into the function V . The a is an effective range 
parameter and will be discussed at a later point. Now T(£) may be written, 



r c * N i 

T(£) = Z X \...\ e V n exp{-0 1 W. o 



where 



-1 * 

+ iX ^ v n W-n } dr. 
30 J 

N J 

x Ii exp{-3.W + i x Jl-V W dr , (11-13) 

1 mO mO m 

m 



1 £-V „V ' . (11-14) 



V = -6.V 0+ i X LVfa 
We now define the functions 

X z U;j) = exp{-6.W j0 + i X I- V W jQ } -1 , (H-15) 

-*. _.->■ -> 

x (£;m) = exp{-3.W + i X A»V W } -1 . (11-16) 

A z 1 mO mO 

T(£) may be rewritten in terms of these new functions: 

1 f f ^0 Nl 

T(£)=z \...\e U n {1 + x U;J)> dr n {1 + x z (^;m)}dr m . (11-17) 
J J j z l J m 2 

The reason for writing T(£) in this form is that we are able to make a cluster 

expansion of the products in the above equation. Performing this expansion, 

we obtain 



V N l N 2 

T(£) - z ] V--\ e °[ 1 + S X z CAjj) + I X z U;m) 

j i m 2 



J-J 



\ N 2 

+ z x z U;j) x z Ui« + 2 x z U;n)x z tt;m) 

i<j 1 1 m<n 2 2 

N l - N 2 - 

+ . z x U;j) x U;m) + ...] ir dr n dr . (ii-is) 

j>m Z]L z 2 j 3 m » 






32 



We now define the functions 






\ 


+ N 2 - 


n 
i=j+: 


dr. II dr 
n=m+l 



(11-19) 



and 



Q. (£) = T. (£) / T_U) ; T n (£) = T (£) . (11-20) 

jm jm U UU 

These definitions allow us to express T(£) in a greatly simplified form. 



1 [1 +N 1 JC 



-> 



T(£) = T Q (£) z [1 + N 1 IQ 10 (A) X z (*;D dr. 

+ N 2 Ui (£)x z 2 a;1) viWyj 



Q 20 W 



XX (t;«X <l;2)dr i dr* + | N 2 (N 2 -l)j J Q 02 (£) x^^Dx^U^dr^ 



+ N 1 N 2J \ Q ll (£)X z (£;1)x z (£;2) dr l dr 2 + ••■ ] ' (11-21)- 

where we have taken advantage of symmetry in order to perform the sums. 
If we define a new function 



J 



h. (£) = \...\ g. U) X U;i).-.x„ (A;j)x. U;D...x_ U;*) 

jm 1 \ 3m z 1 z^ z 2 z 2 

.1 ~ y m ■> 

x II dr. n dr , (11-22) 

. , 1 1 m 

i=l n=l 

where the g's are defined in terms of the O's through an Ursell expansion, 

we may recognize that Equation (11-21) is just the low order terms in an 

expansion of the following expression: 

j m 

TU) = T (£) z" 1 exp{Z E (-^j- -~ h (£) )} . (11-23) 

j m 

From Equations (11-13) and (11-14) we may identify 






33 



n i n 2 



z = T.(0) exp {I E (—- H^ h - C0))> . (11-24) 

U .Tim! ii 

J m J 



The final result for T(|) is given by 



n i n 2 



T(£) - [T (£)/T (0)] exp {E E (-rf- ~~ [h. (£) - h. (0)])} . (11-25) 
U U ."Jim! im p 

j m 



Introduction of Collective Coordi nates'" 

In this section we proceed with the definition of the collective 

coordinates. A transformation to collective coordinates will allow us 

to approximately perform the multi-dimensional integrations appearing in 

the previous section. We may write the total potential energy V in terms 

of its Fourier series: 

->■->- -> -> 

/7tA 2 , 2 2 -ik-r. . 2 2 -ik-r 

v = 4 IIA E .I y, z i e e ij z„e e mn 

V k I.,. ^ + — ^ 



i5 ^ j (kA) 2 + 1 2 Wn (k^) 2 + 1 

■> -> -»•->■ ->■ ->- 

2 -ik-r. 2 -ik.r.„ 2 -ik-r 

z z e e jm Xz e e ^ jO Xz.ee m0 

+ S - + E I + E -1 ] . (11-26) 

j.,m (kX) + 1 j (kX) Z + 1 m (kX) + 1 

The last two terms represent the central interaction terms. The prime 

on the sum over k means that the k=0 term is to be excluded. This 

exclusion ensures that the condition of charge neutrality for the plasma 

is satisfied. 

Now consider the following expression: 

-ik* r , 



E * -iL = y 1 

k 



-*■ -»■ ■+ •+ 



(kx)2 + i = I oDo? + i [cos (k ' r ij } - 1 sln (k " r ij ] 



I ( kA )2 + I cos (k.r..) - (H-27) 



*The collective coordinate technique we present here follows closely 
the development given in References 10 > 23 , 24 , and 25 . 






34 

The second line results from the fact that the sine is an odd function. 

A trigonometric identity now gives 

-> -*- 

- ik • r . . ■* -*■ -> -> 

y e ij y 1 

TtTvV) , -i ~ n [ cos (k-r.) cos (k-r.)] 

k (kX)-^ + i k (u) 2 + 1 ± J j' J 

■+■ ■* -s- -> 

+ E 5 [sin (k-r.) sin (k-r.)] . (11-28) 

k (kX) +1 X J 

The above equation is equivalent to 

-ik-r.. cos(k.r .)cos(k.r .) sin(k. r .) sin(k.r .) 

E £ 1 J ■= 2 E 1 J- +2 E 1 L (II-29) 

(kX) +1 k >0 (kX) +1 k <0 (kX) + 1 

z — Z" 

We separate this expression into k > and k < contributions in the 

anticipation of the definition of the collective coordinates. We now 

define 





r * * 




-»- ->- 


j cos (k- r) 


; k > 
z — 


S(k-r) = 


" 






/ sin (k'r) 


; k < 
z 



(11-30) 



With this definition Equation (11-28) becomes 
-> -»• ■*■->-*■-> 

e~ lk * r n 2S(k-r )S(k-r.) 
E __J_ = j- Ji 1_ ( (11-31) 

k (kX)" +1 k (kX) + 1 

The total interaction energy V may now be written in terms of the 

new coordinates S: 

9 N 2 2 

, , I ., 1 zz n e 

V=^~ f E' \ [ E -±- S.S. 

V 2 k (kX) 2 + l i,j ° ** 

1 ,22 N . 2 H 2 

2 2z e 1 4xz.e 2 4)(Z„e 

+ E S S + E * S.S n + E S S 

amn. aiO a m 

m,n i m 

, 2 ,22 ,22 

4z z e It. e 2z e 

+ E — S.S - — = — B, - — N„ , (11-32) 

a i m a 1 a 2 



35 



where 



2 2 2 2 
a = N^e + N 2 z 2 e 



(11-33) 



The last two terms in Equation (11-32) are needed to subtract out the self 
energy terms (i-j and m-n) which are included in the first two terms. 



Now we define the collective coordinates X: 

p 



X i k = Z 
i,k . 



X 2,k = E 
m 



. 2 2' 
2z e 



<* 3 



9 2 2 

2z 2 e 

— - — S 

a m 



(11-34) 



(H-35) 



The interaction energy V becomes 



9 , 2 
v = 2itX ° E i 



k (kA) 2 + 1 



2 2 

tX l,k + X 2,k + 2X l,k X 2,k 



+ 2 



^4~ S (X l,k + X 2,k> - 2] 



(11-36) 



The constants in front of this expression may be reduced to the following: 

(H-37) 



2 2 

„ ,2 6 z, + Rz„ 

2ttA a e . 1 J?, 



v 



2 z + Rz 
1 * 



where 



n 2 
R = — and 9 = k T 

n e B e 



After defining 



\ 



(kA) 2 + 1 



Y k " X l,k + X 2,k 



we obtain a more simplified form for V 



2 2 

9 z + Rz„ „ 

2 z, + Rz„ , Tc k 
1 2 k 



2 2 

2x e 



k >0 
z— 



' Vi 



Y, - 



2E/L} 



(11-38) 



36 



The second term is summed over k. > because 

z — 



s o = 



k > 
z — 



; k < 
z 



We have manipulated the interaction energy V into a quadratic form 
involving the coordinates Y . The goal of this procedure is to transform 
the multi-dimensional integrals over spatial coordinates appearing in 
Equation (11-22) into multi-dimensional integrals over the coordiates Y, . 

The Collective Coordinate Calculation 



We repeat the definitions of W.„, W „, and V : 

jO mO 

2 
X z i e /i 
W. =-^e- ar j0 /A ; (11-10) 

3 r j0 

2 
XZ ? e -ar A 

\o - "TV e m0 ; (II - n) 

mO 



V-V +EW +ZW . (11-12) 

J m 

The adjustable parameter 'a is well discussed in the literature. °' 24 ' 25 

We will return to the discussion of this parameter at a later point. 

From an inspection of the above equations, however, we may infer that a 

essentially measures the contribution of long-range central interactions to 

the term V_. Proceeding in the same manner as in the previous section, 

we write 

4tta E 2 e tO , ^ . 



e [- 

e z 



2 2 
. + Rz 2 _ 



2xz i e 



1 KZ 2 k >0 (k*r + o 
z— 



S. 
J 



37 



(11-40) 



or 



E W 



JO 



Also we have 



2 , „ 2 

z + Rz 

e z. + Rz„ 




l,k 



k >0 (kA) 2 + a 2 
z~ 



lO m0 
3 m 



2 2 
z^ + Rz^ 

) [— -1 

e L z + Rz J 



9 2 2 

2 X e 



k >0 (kA) 2 + a 2 



(11-41) 



(11-42) 



The net result of these steps Is that we now are able to write down 



an explicit expression for the quantity V 







V - V - E W - E W 

t0 mO 

J m 



2 2 
9 e Z l + Rz 2 2 

f [^ThT 1 { ? \ \ + 2 




/ >n f k (a) \ Y k- 2 jV, (11-43) 
k >U k 

z— 



where 



f k (a) 



a 2 -l 



2 2 
(kA) + a 



(11-44) 



We now need to compute the gradient of V . Recall from Equations (11-13 
and (11-14) that this operation produces the electric field terms appearing 
in the original definition of T(£) . Furthermore, only the central inter- 
action terms contribute to the electric field at the site of the radiator. 
By inspecting Equation (11-38) and (11-42) we see that central interactions 
are contained only in the second term of Equation (11-43). Following 
O'Brien and Hooper we obtain 



2 2 

z + Rz 

v o v o " " e e t-TTzT 1 



- 2 2 
2x e 



„ z n f k (a) Vk k 

k <0 



(11-45) 



We now have sufficient definitions to enable us to calculate T„(£)/T (0). 



T Q U)/T (0) 



M 



e II 



N 2 

dr. n dr 

1 m 

m 



38 



(11-46) 



H : ° 



', :>o) i 
n 
j 



N 

■+■ ^ -*• 

dr . n dr 

3 m 



J- -J 



N_ N 

1 -v 2 -y 



exp{-3 V + i( X e) 1*7X}H dr n dr^ 

3 



m 



J-S 



N. 



N, 



'1 ■* 2 -> 
exp{-3.V„> n dr. n dr 



i 



m 



m 



I 



iexp{-|uE t\\ 2 + 2b k (£)Y k ]} J k ; dY k 
J k k 



[\ «*<" 1 U £ f\\ 2 + 2b k (0) \ ]} J k I d \ 



(11-47) 



where 



2 2 
8 z, + Rz„ 
e r _l 2, 

u = ?; [ Zl + rz 9 ] 

i 1 2 



(11-48) 



and 



b k U) - 



9 2 2 



f k^\ X 



1 ; k >0 



ie.(xe) i'k ; k<0 
i z— 



b fc (£=0) = 



, 2 2 
2 X g 



f k (a) \ 



k >0 



; k <0 



(11-49) 



We have cancelled the self energy terms from the numerator and denominator 



in Equation (11-47) . Here T:' is the' Jacob iari" of the: transformation from the 
spatial coordinates in the collective coordinates. 
Defining A" = uA^ and b' = ub , we may write 






39 



T (£)/T Q (0) = — - . (11-50) 

exp{- - £ [A^Y k 2 + 2b^(0)Y k ]} J k n dY k 



2 
In Reference 23 the Jacobian is shown to be exp{- 1/2 Y } plus small 

correction terms. This reference gives a very thorough discussion of the 

Jacobian transformation and the collective coordinate method. In the 

present work Jacobian corrections are neglected. We give here a general 

result for integrals of the type found in Equation (11-50) : 

/ [VU)] 2 
exp {^ I -JE- } 



T n U)/T n (0) = " k X + K 



X K ' — . (II-51) 

exp {— E ] 

k 1-+AJ 

We at once see that the terms for k >_ give equal contributions in the 
numerator and denominator. These terms thus cancel each other. Also, for 
k < the denominator goes to unity. We have 

j tb'U)] 2 
T U)/T (0) = exp {- E K } , (H-52) 

U ^k<0 1 + A k 

z 



2 2 2 2 2 2 

0/ 7, + Rz^ [f (a)]\ Z (£-k) Z 

- exp {- -J- I 1 2 ] E k 1+ * > . (H-53) 

a z x + Kz 2 k <0 L \ 
z 

This expression may be evaluated by converting the sum over k into an 
integral. We obtain the following result: 

T (£)/T Q (0) = exp {- Y L 2 } , (11-54) 

where y is given by 

y =f T 1 ^ - U +u)]~ 2 B , (11-55) 

e 

and 



B = a 5 u + 2 [1 - (1 + u) 3/2 ] a 4 + [2u + u 2 ] a 3 



4 (1 + u) [1 - (1 + u) 1/2 ] a 2 - 3 [u + u 2 ]a 



40 



2 ' 3/2 
+2 [(1 + u) - (1 + u) ' : 



(11-56) 



We have 



9 z 2 + Rz 2 n 

E r x ^ i . -n M £ 

0. L Zl + Rz„ J ' n, 

i 1 2 1 



) = kT ; 9 . = kT . ; 
e e x l 



4 Tr 3 "0 e 

3 n r n - 1 ; a = -y ; e Q = — 

r o 



' L = V 



We now turn to the problem of calculating the Q. (A) functions which 
are defined in Equations (11-19) and (II-2.0) . A general form for these 
functions is given in Equation (2-56) of Reference 23 : 

2 



Q. U) = 



,,-j-m r 1 „ y k \ 

exp { " 2 t TTa; 

k k 



- E K K ; } 
k 1 + \ 



u- j_m \...\ exp{4 Z y k A k 

J J k 1 + A/ 



(11-57 



, . , , j -> m ■* 
E y u k } n dr. n dr 
k 1 + AT i=l 1 n=l 



where y, is a new collective coordinate given by 



y k = a l,k + a 2,k 



and 



l l,k 



2z e j + -► 

— - — I S(k-r.) 
1=1 



2,k 



1 

n 2 2 

2z e m ■+•-*■ 
— — Z S(k-r n ) . (H-58) 
n=l 



First we consider Q (I): 



V exp -: • ^ E 



! . y ik 



y k b k 



Q 10 (*> - 



2 : 1 + A," , 1 + A? 
k k k \ 



jexp {""J =T 



k A k 



+ 



S 



, E TTJr } dr i 



(11-59) 



41 

2 
The term containing y, , when summed over all k, is proportional to 

Sin (k- r ) + cos (k-r ) = 1. Therefore, this term is independent of 
r and may be cancelled between the numerator and denominator. 

V exp {- E 3-™ > 
Q 10 U) = -— ; T ^ (11-60) 



-if ( - y k b k , .+ 

u j exp (-^rr^> dr i 



Consider now the sum in the numerator. 

2 2 
2z e "*" "*■ 
t = E 1 S(k-r) W£l 

1 " I ° ^K k< ' ' 

k [(kX) + (1 + u)] [(kX) + a ] 



1 ; k > 
z — 



i( X e) 1 e.l-k ; k <0 



(11-61) 



This sum is converted into an integral with r chosen as the polar axis. 
We obtain 

I = z S(x) + i z Lq(x) cos 6 , (11-62) 



where 



6 2 2 1 ,_ ,1/2 

9 3x ^~2 ' [e - e ] , (11-63) 

i a -(1 + u) 

an r a -1 , , 1 , -aax -(1 + u) ax, 
q(x) = - [— ] [— - (e - e ) 

a -(1 + u) x 

, a , -aax ,. , ,1/2 -(1 + u) . ax,-, , TT ,. 

+ — (ae - (1 + u) e ) ] , (11-64 

x 






42 
where x = r/r Q and 6 £ is the angle between £ and r. By examining Equation 
(11-62) we see that I will vanish as x tends to infinity. The result of 
this is that if we take the limit where v-*», N^«>, N/u = constant (thermo- 
dynamic limit), the denominator in Equation (11-60) tends to unity. We 
have 

Q 10 U) = V exp { Z;L S(x) + i Z;L Lq(x) cos 6^} . (11-65) 

Similarly, 

Q 01 U) = V exp (z 2 S(x) + iz 2 Lq(x) cos £ } . (11-66) 

We now may write down several more definitions in terms of these coordinates: 

a tt o~l 2 1 -ar._/A 
-g.W - -6 ± X z x e — e j0 

JO 

= -g Zl W(x) , (11-67) 

where 

e 2 

n T7 / \ e a -aax 
-6W(x) = -x — -^ e . (11-68) 

i 
Also 

- 3 i W mO = _ez 2 W(X) ' ( XI - 6 9) 

The gradient terms are given by 

-3- -> 
_1 ~ aX 

i(xe) £ - v Q W j0 - iLz 1 cos 6 £ {^ — (1 + aax)} , (11-70) 

x 

i "*■ "* -aax 

i( X e) «--V W m0 = iLz 2 cos £ {^y- (1 + aax)} . (11-71 

x 

In terms of dimensionless variables 






43 

n, dr = I x 2 dx^ , (11-72) 

1 z, + Rz n 47T ' 



j 3R 2 df2 , _„. 

n„ dr = — — — x dx 7— . (11-73) 

2 z, + Rz„ 4ir 



By inserting the above expressions into either Equation (11-21) or 
Equation (11-22) we may calculate I _(ft). 

I 1Q (£) = n ± [h 1() (i) - h 1Q (0)] . (11-74) 

After performing the angular integration and rearranging terms we obtain, 

CO 

1 C -5 0/ \ o nr \ sin(Lz.G(x)) 

t r -\ 3 \ 2 z-,S(x) r -3z _W(x) 1 n 

i in U = T~T; — \dxxe- 1 - le 1 [ — 1 — ^rr\ 1J 

10 z, + Rz J Lz.G(x) 



sin(Lz q(x) 
~ [ -l^qTxT-- 1]} ' (II ~ 75) 

For 1^- (£) the analogous procedure gives 



3r f , : 

: 1 + Rz 2 ^ 



0/ n o tt/ \ sin(Lz G(x) 

t c»\ 3R 12 ZoS(x) , -Bz W(x) 2 . , 

X 01 C£) " z, + Rz \ dxx e 2 {e 2 [ LZ2G(X) U 



sin(Lz q(x) 
• 2 

■ Lz q(x) 



-1]} , (11-76) 



where 



-aax 
G(x) = q(x) + ^2~ (1 + aax) . (11-77) 

x 

We now turn to the calculation of the second-order terms Q (ft) , 
Q (ft), and Q (ft). Corresponding to these, we have the following three 
cases. 






44 



Case (1) : y. = 



2e 



z E S(k-r.) 



(II-78a) 



Case (2) 



y k = 



2e 



2 -> ■+ 

z„ S S(k-r ) 
2 ., m 
n=l 



(II-78b) 



Case (3) 



2e 



{ Zl S(k- ri ) + z 2 S(k-r 2 )} 



(II-78c) 



The collective coordinate evaluation of the second sum in the numerator of 
Equation (11-57) proceeds exactly as in the evaluation of I in (&). We may 
immediately write down the following results. 

Case (1): - E , , A . " z {S(x ) + iLq(x ) cos 6 } 
k i + A k i X 1 L 



+ z 1 -{S(x 2 ) + iLq(x 2 ) cos 6^ 



(II-79a) 



Case (2): - I r + ; = z {S(x ) + iLq(x ) cos 6 } 
k k 



+ z 2 (S(x 2 ) + iLq(x 2 ) cos 6 } , 



(II-79b) 



Case (3): - £ 1 + = z {S(x ) + lLqCXj) cos e i } 

k k 



+ z 2 (S(x 2 ) + iLq(x 2 ) cos S^ 
2,. 



(II-79c) 



Next we need to evaluate the terms - k k. . We consider y and 

1 + AT 



concentrate on Case (1) : 



-V -V -»--»- 



2 •*■ ■> 
y 2 -— z 2 {S 2 (k-r 1 ) + 2S(k-r 1 )S(k.r 2 ) + S 2 (k-r 2 )} . (11-80) 






45 

2 
The terms in S , when summed over all k, are independent of coordinates 

as before. These terms, therefore, will cancel between the numerator 

and denominator in Equation (11-57). The remaining contribution from 

2 • • u 
y is given by 

y k = ^~ z 1 SCk-r^ S(k-r 2 ) , (11-81) 

-> -> -> -> 
2e 2 2 S(k-r 1 )S(k-r 2 ) 




Z l ? _ 1 + A^ K. 



k 



->■->•■-*■ 



2e 2 2 cos[k-(r 1 -r 2 ] 

= " ~TT Z l U Z 5 • (11-82) 

k ( k x) z + (1 + u ) 

Equation (11-82) results from the application of a standard trigonometric 

identity. The sum over k is converted into an integral with r „ as the 

polar axis. We obtain 

l r y ik 2 9 e a 2 - (1 + u) 1/2 . 

- 2 1 rnj = - z H3^ e 12 ■ (II - 83a > 

Similarly, 

2 

t y.A. e 2 1/2 

r,o Q M\. X v k k 2 e a -(1 + u) ax,„ ,__ _„. 
C ase (2 ). - 2 I —- = -^ — _ e 12 . (n-83b) 



i yfv 9 2 n j. ^/ 2 

r^o~ ^-^. - v k Tc e a -(1 + u) ax,„ ,__ „_ x 

Case (3): - - I 3-—, = - 8 , — — - e 12 . (II-83c) 

K K 1 12 



Now that we have obtained the form of the expressions needed to 
evaluate the second order terms, we see that the integrals in the 
denominator of Equation (11-56) once more tend to unity when the thermo- 
dynamic limit is taken. Also we may identify factors appearing in the 
second order result to be just the Q-, n (&) and Q (£) that we evaluated 
earlier. We may then write 






46 

an 1/9 

Q 20 (A;1,2) = Q 10 a;l)Q 10 U;2) exp {-z^ -1 |__ e ~^ + u ) ax 12},(II-84a) 

i 12 



ft 9 1/9 

Q 02 (£;l,2) = Q 01 (£;1)Q 01 (£;2) exp {-z\ -~ |^— e" (1 + u) ax 12}, (II-84b) 

i 12 

ft 2 1/9 

Q n (£;l,2) - Q 10 (£;1)Q 01 (£;2) exp {-z * 2 ^ ^— e~ (1 + u) ax 12}.(Il-84c) 

i 12 

We define the function I (£) by 

I 20 U) = \ nj [h 2Q (£) - h 2Q (0)] , (11-85) 

where h„_(£) is given in terms of the Q. functions by 
20 jm 



2 \^\ il - 



h 2Q (£) .. v Y r 1 \ dr 2 ^Q 20 (£;1 ' 2) " Q 10 (A;DQ 10 (Ji;2)} x z d;Dx z (t}2), 



j4j ( 



(11-86) 



U 2 (aA) 6 j.i dx 2 Q 10 (£;l)Q 10 (£;2)x z U;l)x z U 



„ 2 _, .1/2 

x {exp [-< ^| e U + u; aX 12] -1} . (11-87) 

19. 3x, „ 
1 12 

The other expressions for h. (£) in second order are very similar so that 

we concentrate here on h„ n (£). Define now 

D 12 (x 12 ) - exp (-4 £ §i- e"» + ») 1/2 -12) -1 . (11-88) 

1 12 

We are free to make the following expansion 



D 12 (x 12 ) -I (2k + 1) V k (x 1? x 2 ) P k (cos 6 12 ) , (11-89) 
k=0 

where P, is the Legendre polynomial and 8 „ is the angle between x and 

■> 
and x . V (x ,x ) is given by 



ir 
-v -> 



V k (x r x 2 ) »| J D 12 (x 12 )P k (cOS 6) Sin 6 d9 • (H-90) 

o 



47 



Inserting Equation (11-89) into Equation (11-87) we obtain 



6! w 



h 2Q U) = (aA) E (2k + 1) \d Xl \dx 2 ffc^) f (x 2 ) x 

K— U 



X 



\ (x l' x 2 ) P k (cOS 9 12 } ' , (H-91) 

where 



-»■ 



f(x) = exp {z 1 S(x) + iLz q(x) cos 9}x 



x {exp [-B Z;L W(x) + iL Zl VW(x) cos 6] -1} . (11-92) 

P, (cos ) may be written 

P k (cos 6 12 ) = J q £m [{^g| P » (cos 8l ) p£ (cos 2 )cos[ m ( W ] ( 

(11-93) 

where 

1 ; m=0 

e = 
m 

2 ; m^O 

If we insert this expression into Equation (11-91) we find that the only 
non-vanishing contribution comes from the m=0 term. Thus the expression 



for h 20 (£) may now be written 



h 2Q (£) = (aX) 6 E (2k + 1) \ dx ± \dx 2 f( Xl ) f(x 2 ) 



\ dx l 1 dx 2 f ^ x i' 



x V k ( X;L ,x 2 ) P k (cos 8 ) P k (cos 9 ) . (11-94) 

The angular integrals may now be performed with the aid of the following 



identity: 



i 



(i) k 4tt j k (A) = Vdn e iAc0s6 P k (cos6) , (II-95) 






48 
where j, is the spherical Bessel function. After the angular integrals 



are performed h„„(£) becomes, 



h 20 (£) = (aA) 6 E (2k + 1) (-l) k I d X] x 2 ldx^ »' 



\^Ay 



x F 1 (£;l) F 1 (£;2) V k (x^) , (11-96) 



where 



1 (£;1) = e*l° v *l' (e P'i"^i' j k (Lz 1 G(x 1 )) - j^Lz^x^) }, (11-97) 

F 1 (0;1) = e Z l S(x l } {e" 3z l W(x l ) -1} 5 k Q . (11-98) 

At this point V (x ,x„) remains unspecified. We introduce here a linearized 
approximation to Equation (11-88) : 

Now V (x ,x ) is given by 

CO - ,„ 

,-. - 9 6 2 f -(1 + u) x/ ax 

V k (L) (x r x 2 ) - - \ z\ f i- \ S — ^ P k (cos e)sine d6 .(11-100) 

l o 12 

27 
This integral has been evaluated by Swiatecki with the following result: 

V < L >(x x ) = -z 2 ^ \ + l/2 (a ' V \ + l/2 (a ' V 

\ t x i'V z i e. 3 I s K } 

1 J x x x 2 ' X l X 2 . 

1/2 
where a' = (1 + u) a. 

When we insert this result into Equation (11-96) , we obtain a final form 
of I 20 U). 

1 20 (L) = -z 2 ^ V-^\^f 3a 2 I (2k + 1) (-l) k {20} , (11-102) 
x 1 2 k 






00 

{20} = f x^ 2 I k + 1/2 (a' x 2 ) e Z l S(x 2 ) [e" Bz l W 10 (x 2 ) j^LGC^)) 
o 

CO 



49 



x. 



x[e Bz l W 10 (x l } J k (z l LG(x 1 )) - J k (z 1 Lq(x 1 ))] chyh^ 



oo 

~ 6 k,0 J x 2 I l/2 (a X 2 } e X 2 [e 1 10 2 - 1 - 



X l /2 K l/2 (a ' X l } eZlS(Xl) C e BZ 1 W 10 (X 1 } -1] d Xl dx 2 .(11-103) 



CO 

J 

X 2 
Note that a factor of 2 has been cancelled and the integration now covers 
one-half of the positive quadrant. This results from the fact that the 
integrand is symmetric under the interchange X- ■*•*■ x„. In a similar 
manner we obtain 

I 02 (L) = -z 2 ~ [ - I Rz ] 2 3a 2 I (2k + 1) (-l) k {02} , (11-104) 
i 1 2 k 

CO 

{02} = f x 2 /2 I k + 1/2 (a' x 2 ) e Z 2 S(x 2 ) [e" 6z 2 W 10 (x 2 ) J k (z 2 LG(x 2 )) 



[ 3 / 2 v 
J X l \ + 



-i k ( Z2 Lq(x 2 ))] J X — ^ + 1/2 (a' x x ) e Z 2 S(x l^ x 



X 2 



[e 6Z 2 W 10 (X 1 } J k (z 2 LG(x 1 )) - j^z^q^) ) ] dx^ 



CO 

,0 J xf Il; 



~ 6 k,0 \ x^ /; I l/2 (.- x 2 ) e*2 Sfa 2 ) [e" S *2 W 10 h 2 ) -l] 






50 



x 

X 



oo 

\ X f 2K l/2 (a ' V e z 2 S(x l>t e " 3Z2Wl ° CXl) -^ dx l dx 2 • t 11 " 105 ) 



The symmetry mentioned above is not present in I (L) . 

a 

z ii (L) = " z i z 2 ~f~ [ z + Rz ]2 R 3a2 z < 2k + D(-D k Ui) , (11-106) 

i 1 2 k 



{11} - f 4 /2 L + 1/9 (a' xj e 2 2 S(x 2 ) [e- Bz 2 W 10 (x 2 ) Jl (z LG(x 9 )) 



5 



2 "k + 1/2 W *2' LC ' iU * J k vz 2^ x 2' 



00 



-J k (z 2 L q( x 2 ))] 1 xj[ /2 ^ + 1/2 (.' x x ) e 2 l S ^l> 



X 2 



x[e 3Z 1 W 10 (X 1 } J k (z 1 LG(x 1 )) - J k (z 1 Lq(x 1 ))] dx^ 

CO 

] X 2 /2 \ + l/2 (a ' X 2 } eZ 2 S(X2) [e _Bz 2 W 10 (x 2 ) j^LG^)) 



+ \ x 



*2 



, M MH ( 3/2 T , f , Zl S( Xl ) 

-J k ( Z2 Lq(x 2 ))] ^ X]L I k+1/2 (a' X] _) e 1 l' 



x[e 6Z 1 W 10 (X 1 } J k (z 1 LG(x 1 )) - J k (z 1 Lq(x 1 ))] dx^ 



CO 

:,0 j x^ /2 1^ 



" 6 k,0 \ 4 /2 I 1/2 (a' x 2 ) e z 2 S(x 2> [e'^W-l] 



oo 

v \ 3/2 , t , z 1 S(x 1 ) r -6z 1 W irv (x n ) , -, , j 

x \ x K , (a 1 x ) e 1 1 [e 1 10 1-1] dx dx ? 



x 2 






51 



00 

I 



~\,0 \ 4^ K l/2 (a ' x 2 } «"2 S( ? 6 2 ) [«" 8l 2 W 10 (x 2 ) -: 



x f 2 x^/ 2 I 1/2 (a' Xl ) • Z l 8(x l ) [.e" B *l W 10 (x l ) -l] d X] _dx 2 . (11-107) 

We should mention here that the linearization introduced in Equation 
(11-99) is not forced upon us. In a more general nonlinearized treatment, 
the integral in Equation (11-90) can be evaluated numerically. In fact 
the computer program needed for the nonlinear calculation has already 
been developed. However, it requires considerably more CPU time than the 
linearized version. We employ the linearized version because it is 
quicker and, in our cases, yields results in agreement with the more 
general nonlinear version. 

Now let us define 



I^L) = I 10 (L) + I Q1 (L) 



I 2 (L) = I 2Q (L) + I Q2 (L) + I U (L) 

T X (L) = exp {- Y L 2 + I 1 (L)} y 

T 2 (L) = exp {- Y L 2 + I (L) + I 2 (L)} . 

T (L) is defined as the "first approximation" to T(L) while T„(L) is 
called the "second approximation." 

Asymptotic Microfield Distribution Function 

In order to obtain a microfield distribution function, we must 
perform the following numerical sine transform. 






52 



-S 



P(e) = 2ett ' J dL LT(L) sin (eL) . (11-108) 

o 

There are many techniques for computing this transform which yield 

accurate results for small values of £ (generally for e<10, in units 

of e . ) However, as the value of e increases, there is a loss of 

numerical precision which is characteristic of integral transforms of 

this type. In order to extend P(e) to large values of e we must develop 

an asymptotic expression. This problem is discussed by several 

.« I,10j23,28 TT . , _ . , 
authors. We wish to outline the treatment of this subject 

given in Reference 10 as it is particularly suited for this theory. 

The model for this asymptotic expression is known as the nearest- 
neighbor approximation (NNA) . This approximation states that high fields 
are produced by a single ion perturber during a close encounter with the 
radiator. This model assumes that for sufficiently large values of e, 
the probability that two or more perturbing ions contribute to the field 
is essentially zero. The probability that a single ion produces an 
electric field e is related to the probability of a close encounter with 
the radiator in the following manner: 

2 
P 1 (e 1 )de 1 = 4Tir 1 g^ (r^ d^ 

■* 2 

x g (x ) dx (11-109) 



z, + p,z„ i 6 i v r 1 



P 2 (e 2 )de 2 = 4Trr 2 g 2 (r 2 ) dr 2 



3R 2 

x -> g (x ) dx„ . (11-110) 



z + Rz 2 2 6 2 v V 2 

The g's are pair correlation functions, which we choose to be the non- 
linear Debye-Huckel expressions. 



53 

2 9 f 1/2 

r \ r a e -\l + u; ax,, ,„,. ,,_ . 

g 1 (x 1 ) = exp {- X z 1 -^- -- e 1} f (11-111) 

1 i 

2 6 . , 1/2 
g 2 (x 2 ) - exp {- X z 2 2^~ g - e 2} . (11-112) 

1 i 

In terms of dimensionless variables 

e ± = — (1 + axj) e dX l , (11-113) 

X I ' 

Z 2 
£ 2 = ~2 (1 + ax 2 ) e & * 2 * (II-114) 

x 2 

When these expressions are differentiated and inserted into Equations 
(11-109) and (11-110) we obtain the following expressions for the 

asymptotic microfield distribution functions. 

/ e 2 ft j. >!/2 

4 r ea -(1+u; ax, , 

3x exp {- X z — — — e 1} 

P ( e ) - r 1 1 - 1 - 1 - 9 i 3x i 

i; E r L z + Rz 2 J — X — — , (11-115) 

z..e 1 {2 + 2/ax 1 + ax } 
/ 8 2 /I JL ^/ 2 

•,4 , ea -(1 + u) ax--, 
3x 2 exp {- X z 2 — 3— e 2} 

r 1 ? 

P 2 (£ 2 ) =[ Z;L + Rz 2 ] z 2 e~ aX 2 {2 + 2/ax 2 + a^} ' (II " 116) 



P Asym (£) = W + P 2 (£ 2 ) ■ (II ~ 117) 

The present method of joining the computed P(e) and the asymptotic P(e), 
which differs considerably from that appearing in the above reference, is 
considered in detail in an appendix. 



SECTION III 



DISCUSSION OF THE RESULTS 



Introduction 

In this section we present graphical results for the microfield 
distribution functions and x-ray line profiles calculated using the 
theory as given in Sections I and II. Experimental observations of 
Lyman-a for Ne X have been reported only recently.* However, it has 
been verified that this theory duplicates the results for the micro- 
fields and line profiles given in Reference 23. 

First we present figures demonstrating the behavior of the micro- 
field distribution functions under variations of the different plasma 
parameters. We also point out and discuss trends and important 
features which are exhibited by these curves. Following this, we 
present Stark broadened x-ray line profiles computed using the above 
microfield distributions. General features of these profiles are 
also discussed. 

After presenting the figures we consider the various approximations 
made in the development of the line profile formalism in Section I. 
These approximations determine several basic validity criteria for 
the application of this theory. 



*The observations to which we refer here were made in connection 
with laser- imploded pellet studies at Lawrence Livermore Laboratory. 29 

54 






55 
Electric Microfield Distribution Functions 

In this section we present graphical results for the electric 
microfield distribution functions computed using the theory of Section 
II. The first three figures illustrate the behavior of the microfield 
functions for multiply charged hydrogenic radiators perturbed by singly 
charged ions. The field variable s is in units of z (= e/r 2 ) . a = 
r Q /A D . In Figures 1 and 2, the charge of the radiator is +9 and +17, 
respectively. Figure 3 shows a comparison of the microfield functions 
for different radiator charges at a given value of a.. 

Two effects are immediately obvious from these figures. First, 
for fixed radiator charge x, when the value of a. is increased, each 
successive microfield curve has its peak shifted to lower values of e, 
becomes narrower, and has its maximum value increased. This behavior 
is analogous to that noted previously for x=l- 10 ' 24 ' 25 Second, 
Figure 3 indicates that for fixed a., as the value of x is increased, 
the behavior is similar to that noted above when a. is increased. 

Furthermore, a comparison of Figures 1 and 2 with Figure 4 of 
Reference 24 reveals that as x increases the relative sensitivity to 
changes in a increases. This increased sensitivity is due to the fact 
that in many functions defined in Section II the parameter a is 
multiplied by x- 

Figures 4 through 7 illustrate the behavior of the microfield 

distribution functions for the a = 0.2 and _a = 0.4 cases, corresponding 

to those in Figures 1 and 2, when the parameter T is varied. This is 

R 

the ratio of the electric kinetic temperature to the ion kinetic 
temperature. 

T = kT /kT. = 9/6, . (III-l) 

R e i ex v x ■*•' 



56 

Figures 8 and 9 illustrate the effects of T -variation on microfield 

distribution functions for a = 0.2 and a = 0.4 when hydrogenic 

Aluminum (x *• +12) is perturbed by a plasma containing only A£ XIII 

ions (z = +12) . 
P 

As the value of the parameter T is increased, in all cases the 

R 

microfield distribution function peaks shift to lower values of e. This 
behavior might be anticipated from the form of the following relation: 

a' = (1 + u) 1/2 a, (III-2) 

where, if R = 0.0, 

U = 6 e /e i * z l = T R ' \ ■ ' (IH-3) 

The parameter a*_ appears in several functions in the definitions of both 
I.. (L) and I (L) . Thus, if we regard a_^ as a modified plasma parameter, 

we see that as the value of T is increased, the result is an effective 

R 

increase in the ji value. A direct comparison of Figures 4 through 9 

with Figures 1 and 2 indicates that when the value of T is increased, 

R 

these T -variation curves indeed exhibit behavior similar to that noted 

when the _a value was increased. While the dependence on T is obviously 

R 

strong in all cases considered, the analytic form of this dependence is 

extremely complicated and difficult to assess. 

1/2 
To see more clearly the effect of the (1 + u) factor, let us 

consider the following: 

a' = (1 + u) 1/2 a = (1 + u) 1/2 r Q A D = r /A' D , (III-4) 

where 



2 
x \ = RF 1 - (i + ")] """ . (IH-5) 



r 4irne ,., , _ N1 -l/2 



57 
The definition of u is given below Equation (11-56) ; we now write it in 
the form 

9 e N 1 Z 1 + N 2 Z 2 9 e Vl + n 2 Z 2 

u = t; Vl + n? z ; = e: n — • < II]: - 6 > 

i 11 2 2 i 
The second step in the above equation results from the overall charge 
neutrality of the plasma. Inserting this result into Equation (III-5) 
we obtain 

»- D -[4.. 2 (f- + m A; n 4 r i/2 _ (m . 7) 

e i 

This equation is a generalization of Equation (19) in Reference 30. 
Thus we obtain for a' : 

2 2 
r/ 2 / n n z + n z f 

a = r Q [4iTe (— + Q )] . (III-8) 

e i 

In our computations, we regard T and n (the electron number density) 

as fixed by the parameter a.. This means that, for a given value of a, 

an increase in T is accompanied by a decrease in the ion temperature. 

Equation 8 gives the explicit dependence of a' on the ion temperature. 

In addition this equation suggests that the shift of the microfield 

peaks might be due to increased importance of ion correlations as ion 

temperature is lowered. This correlation effect is especially apparent 

in Figures 7 and 9. Looking at these figures we see. that, as T takes 

R 

higher values (ion temperature decreases) , the A£ microfield distributions 

peak at smaller field values than do the Argon distributions. This is 

to be expected since interparticle correlations are stronger in the 

kl-kl system than in the Ar-H system. On the other hand, as T decreases 

R 

(the ion temperature increases), the figures indicate that the stronger 






58 

fields are more probable in the A£ plasma. This is due to weakened 

correlations; the average interactions in this case will be stronger 

in the A£-A£ system than in the Ar-H system. 

In Figures 10 through 13, we present the results of computations 

in which T R is set equal to unit and the parameter R is varied. Now R is 

the ratio of the density of charge z perturbers to the density of 

charge z^ perturbers: R = 0.0 (°°) .corresponds to the case where the 

ion perturbers are all of charge z^ (z ) . The function u is given by 

2 2 
■ x + Rz 2 

u = Z± + rz 2 (T R = « • (in-9) 

It varies smoothly from a value of z to a value of z as R goes from 
0.0 to ». Because of this we might expect that when the perturbing 
ions have z 1 and z^ values nearly equal, the variation of R causes 
very little change in P(e) . Indeed this is the case for A£ microfields 
when z 1 = +12 and z^ = +11. For this reason we omit figures showing 
the R variation calculations forthe A£ system. 

The most noticeable effect of an R variation occurs when z and 
z 2 are very different (Figures 10-13). Furthermore, it should be noted 
that for very different z ± and z 2 (with z 2 > z ), most of the variation 
of P(e) occurs between the values of R = 0.0 and R = 0.1. This behavior 
is due to the fact that for a situation where z « z and R is only 
slightly greater than zero, most of the free electrons are already 
contributed by perturbers of charge z . This is illustrated by 
considering, for example, the following case for argon (z = +1, z = 
+17): 

R = o.o n. - n n n = 0.0 

1 e 2 

R ■ 0«1 n. = 0.37 n n = 0.037 n 

1 e 2 e 

R = OT l^ - 0.0 n 2 = 0.059 n (111-10) 







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Figure 7. Electric microfield distribution 
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charge of +17. a = 0.4. The ion 
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Figure 9. Electric microfield distribution 
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76 




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85 

In the present work, we consider n to be fixed by the a_ value. Then 
n and n are determined by the requirement of overall charge neutrality 
for the plasma (Equation II-4) . 

For lower a_ values, an increase in R causes the microfield peaks 
to shift to higher field values. At higher a. values, however, the 
increase in R produces a shift to smaller field values. The behavior 
at higher a. values can be explained in terms of particle correlations. 
That is, an increase in the value of R results in an increase in the 
effective plasma parameter a/_. At lower a. values, correlations are 
weaker and the observed shifts indicate that as R increases, the average 
strength of interactions within the system increases. 

Stark Broadened Line Profiles 

In this section we display the broadened line profiles computed 
using the electric microfield distribution functions shown in Figures 
1 through 13. Figure 14 shows the relative contributions to the 
Lyman-a Ne X line profile from the several broadening mechanisms that 
are included in our calculations, together with the combined result. 
The conditions represented in Figure 14 correspond to an a. value of 0.4. 
The relative importance of the Doppler effect is of interest. For the 
temperatures discussed in this paper, the Doppler effect is much more 
significant — compared to the electron-broadening contribution — than 

was the case when dealing with more conventional plasmas (e.g., n = 

17 -3 
10 cm and T = 40 000°K) . The qualitative features of the Stark 

profile are also different: here the electron contribution produces 

a sharp spike which sits on shoulders provided largely by the ions. 

Figure 15 presents a plot for Ar XVIII that is equivalent to Figure 14 



86 
for Ne X. The qualitative information is the same. 

Figure 16 shows for both Ne X and Ar XVIII, families of Lyman-a 
profiles, each of which corresponds to the same T but different n. 
It is evident that as n increases, the profile changes greatly, both 
in shape and width, thereby illustrating the density sensitivity of 
the line profile- The practical sensitivity that might be inferred 
from comparison of experimental and theoretical line shapes depends 
on at least two factors, (1) how much of the line profile can be 
experimentally observed, and (2) the resolution of the x-ray spectro- 
meter used. 

Figure 17 shows for Ne X and Ar XVIII, families of three Lyman-a 
profiles, each of which corresponds to the same density but different 
T. The results here imply that the frequently mentioned insensitivity 
of plasma-broadened line profiles to variations in T is significantly 
reduced as x increases. The fact that Doppler broadening plays a more 
significant role in broadening the argon lines than it does for neon 
is due to the fact that the effect depends not only on mass, but also 
on the radiator z. 

Figures 18 through 20 show the behavior of Lyman-a profiles for 
hydrogenic neon, aluminum, and argon, respectively, under variation of 
the parameter T . The electron density is the same in all these cases. 
For the curves on the left the electron temperature is 1019.2 eV while 
on the right it is 254.8 eV. In the neon and argon cases, the ion 
perturbers have a charge of +1. For the aluminum case the ion perturbers 
have a charge of +12. Three general features of these Lyman-a profiles 
can immediately be noticed. First, since the electron broadening of the 
unshifted central component is not directly affected by the ion 



87 

microf ield distribution, most of the variation of the central component 
observed in these figures is due to variations in the Doppler broadening. 
Second, since the wings of the profile are determined almost entirely 
by the microfield distribution, we immediately see that the variations 
in the wings of the profile reflect the structure of the microfield 
distribution functions discussed earlier. Finally, the line profiles 
show a significant amount of structure in the region of the shoulder. 
The electron broadening of the unshifted component plays an important 
role in producing this structure. To see this more clearly, compare 
the left hand T = 4 curve with the right hand T = 1 curve in each of 
these figures. By doing this we isolate the electron broadening effects 
since the ion kinetic temperatures (i.e. ion broadening and Doppler 
broadening) will be the same for each curve. The electron temperatures 

are the only difference between the two curves. Since electron 

-1/2 
broadening scales as T , the electron broadening is greater in the 

right hand curve. That is, the electrons fill in the shoulder on the 

right, whereas the shoulder of the left hand curve is quite pronounced. 

In addition, there are significant variations in the structure of the 

shoulder within each family of curves. 

Figures 21 through 23 demonstrate the same T -variation calculations 

R 

for the Lyman- 3 profiles of neon, aluminum, and argon. Doppler broadening 
effects are more important for these lines than for Lyman-a due to the 
increase in the unperturbed transition frequency. Once again, Doppler 
broadening is responsible for most of the variations around the central 
dip, within a given family of curves. However, by isolating the 
electron broadening effects as indicated above, we can see that these 
effects also contribute significantly to the structure of the central 
dip. 



50 






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116 
The behavior of Lyman-a profiles for neon and argon under the 
variation of the parameter R is displayed in Figures 24 and 25. 
Figure 24 shows neon and argon profiles for a = 0.2. As we mentioned 
above, most of the variation occurs between R = 0.0 and R = 0.1. 
Notice that the shoulder becomes more pronounced as R increases. The 
reversal shown on the right side of Figure 25 seems to indicate that, 
at higher a values, particle correlations will lead to reduced line 
widths . 

Figures 26 and 27 show Lyman-g profiles for neon and argon under 
the same R variations. The qualitative information is the same as 
for the Lyman-a profiles. 

Validity Criteria for this Theory 

' We pointed out in Section I that the neglect of fine structure 
of the radiator energy levels leads to a criterion for the validity 
of the linear Stark effect in calculating the line profile. The 
requirement that the level shifts due to the Stark effect be greater 
than the fine structure splitting sets a lower limit for the electron 
density. That is, 

n > 2.4 x 10 16 (-V 572 (1 - M 3/2 . (IH-11) 

e n n 

u u 

An upper bound on the particle density may be determined by requiring 

that Stark shifts be smaller than the separation of radiator levels 

of different principal quantum number. This critical density is 

referred to as the Inglis-Teller limit: 

^ in 23 9/2 -15/2 (111-12) 

n < 1.5 x 10 z n . Vi-xj- j-^j 

e u 



117 

This restriction should ensure that the average, broadening interactions 

are weak enough that (1) they may be treated by perturbation theory, 

and (2) the No-Quenching Approximation will be valid. 

The densities employed in the present work fall into the range 

set by the above inequalities. However, there is an additional point 

which must be considered. The dipole approximation to the radiator- 

perturber interaction has been made. Higher multipole moments will 

become important when the radiator-perturber separation approaches the 

radiator size. To minimize possible contributions from the higher 

multipoles we should require r„ > r where 

On 

2 

n a„ 

r = -^ , (111-13) 

n z 

is the radius of the radiator in the excited state of principal quantum 

number n . In the present treatment, our "worst" case for this criterion 

31 
is Lyman-3 for neon with r_ ~ 3r . Bacon " has employed the Classical 

(J n 

Path Impact Theory to treat exactly the effects of higher multipole 
moments on Lyman-3 in hydrogen at a. values (~ .3) close to those in the 
present work. His work indicates that the inclusion of higher multipole 
terms (as well as time ordering effects) tends to fill in the central 
dip of Lyman- 6- Thus, we might expect our widths to agree with Bacon's 
results but it is probable that there would be differences in the line 
center due to our lack of higher multipole moments in the radiator- 
perturber interaction. We mention in passing that it may be possible 
to adjust the correlation cutoff given in Equation (H-9) so that it 
accounts for strong collisions as well as for electron correlations 
(see the discussion of this cutoff given in Appendix H) . 



118 

Whereas when dealing with hydrogen plasmas the plasma parameter 

2 
T = e /rJkT is an indication of the strength of coupling, in plasmas 

containing multiply charged ions this definition must be replaced by 
2 2 

r' "W~ . Ciii-14) 

2 
where <z > = (1 + u) is a generalization of Equation (4) in Reference 

32. In the present work, we require V < 1, in which region we expect 

the microfield calculation to be very reliable. This is equivalent to 

a' < 1.732 . (111-15) 

Two of our cases (a = 0.4, T = 2.0 and T = 4.0 for aluminum) violate 

K. K 

this inequality and must be viewed with uncertainty. Equation (111-15) 

gives the explanation of why we restrict ourselves in this work to the 

consideration of ji values 0.2 and 0.4. 

In Section I we indicated that the Quasi-static Approximation for 

the ions is valid for Ago > oo , T ,. In all of our cases, go , t 

p(Ions) p(Ions) 

falls in the very center of the line profile so that except for a small 
frequency range around the line center, the ions are adequately treated 
as static. The plasma frequency for the electrons is given by 

w / w1 «. s = 2.72 x 10" 12 n 1/2 , (111-16) 

p(Electrons) ' v ' 

in Rydberg units. This frequency falls off our scale in all the figures 

presented so that the electrons cannot be treated as static in the 

frequency range we display. The validity region of the second-order 

perturbation treatment in this work covers the center of the line profile 

and extends roughly out to the electron plasma frequency. The strong 

collisions, which contribute to the line profile for Aw > go ,„„ ., 

p(Electrons) 

are not adequately treated by finite-order pertubation calculations. 



SECTION IV 

CONCLUDING REMARKS 

The goal of this work has been to perform a preliminary study of 
Stark broadening in hot dense plasmas. By "preliminary" we mean that 
the results presented in the previous section should indicate the 
direction which future work on this problem should take. A basic 
conclusion resulting from the present work is as follows. Although 
Doppler broadening generally dominates in the center of the line 
profiles, the structure near the center (i.e. the shoulder of. Lyman-a 
and the dip of Lyman-3) shows significant sensitivity to density- 
temperature variations. A large part of this structure is produced 
by the electron perturbers. For the cases where ions and electrons 
have different kinetic temperatures, the electric microfield 
distribution functions produce significant effects on the line shape. 
Since these effects can generally be discussed in terms of particle . 
correlations, they appear to indicate that correlations will play an 
important role at higher densities. In addition, at low a values there 
is an indicated sensitivity to small concentrations of high-z ion 
perturbers . 

One immediate goal of future research is the extension of line 

25 

broadening theories to higher densities, vL0 (e.g. densities expected 

in controlled fusion experiments, which are significantly greater 



119 



120 
than those employed in the present study) . We can expect that corre- 
lation effects, quantum degeneracy effects, and strong-collision effects 
will become increasingly important at higher densities. These effects 
must be properly included in subsequent line-broadening theories. 

A recent experimental measurement of the Lyman-a profile for 

33 

hydrogen at moderate plasma density and temperature J produced an 

experimental- theoretical discrepancy. The measured shape of the line 

34 
center was about 2.5 times wider than theoretically predicted... Griem 

has conjectured that this discrepancy results from an additional 

broadening mechanism: the screening of ions by electrons is dynamic 

rather than static and hence gives rise to density fluctuations in the 

screening electron density, which in turn provides additional line 

broadening. This possible explanation deserves further study since it 

could be important in analysis of laser implosion experiments through 

the use of x-ray line profiles. 

35 
Work in progress at the University of Florida is aimed at 

development of a line broadening theory capable of treating all of the 

above effects. Particular emphasis will be given to the study of line 

profiles emitted by radiators immersed in dense, highly ionized systems. 



APPENDICES 



APPENDIX A 

THE INTERACTION V 

er 

In the model for our plasma as discussed in Section I, we have a 

hydrogenic radiator fixed at the origin of coordinates. The ionic 

radiator has a nuclear charge of z. Using bare Coulomb interactions, 

we may write the interaction between the radiator and one perturbing 

electron as follows: 

2 2 
er R - r . 1 r . 

The first term corresponds to the Coulomb interaction between the i-th 
perturbing electron and the single bound radiator electron (located at 
the position specified by R) . The second term is the interaction between 
the radiator nucleus and the i-th electron. Let us expand the first 
term in the usual spherical harmonics, 

V er " / JITTT) -TTT Y £m*^ Y £m ( ^ " Z ^ > (A " 2) 

£,m r ' i 1 

> ' l 1 £=1 m r 

> 

We now introduce two approximations which allow us to simplify the 

above expression. The first step is to retain only the £=1 term in the 

sum of Equation (A-3) . This procedure is referred to in the literature 

as the Dipole Approximation to the radiator-perturber interaction. Next, 

we take r = |r.|. This approximation neglects the possibility of 

"penetration of the radiator" by the perturbers. The validity of these 

two approximations in the classical path theories is assured by strong- 

12 4 
collision-cutoffs in the classical averages. Validity criteria 

122 



123 
pertaining to the present work are discussed in Section III. Now we 
may write 



2 2 1.2,+ 

„i e e , _ 4n-e R 



er r . r . , ' 3 r\ 
1 x 1 ' x 1 m=-l ' x 



7 Y lm* ( V Y lm ( V ' (A " 4) 



2 1 / 2 Id 

e , _ 4ire R 



= -X IITT + ^ 



x ' m=-l ' x 



? V^t Co ) , (A-5) 



where x = z ~l is the net charge of the radiating ion. The second term 
may be recognized as a vector dot product. 

V* = -X y^ + eg • !(?.) , (A-6) 

where 

->- 
r . 

^ (? i } = e ^V • < A - ?) 

Since the perturbing electrons remain "outside" of the radiator, 

they "see" a radiator of net charge X- This suggests that we may treat 

the first term of Equation (A-6) exactly by selecting as a basis set 

for the perturbers the free particle Coulomb wave functions for electrons 

moving in the field of an ion of charge X . With this choice of perturber 

wave functions, we redefine V : 

er 

V 1 = eR • E.(r.) , (A-8) 

er x x 

-> •+ 

where E.(r.) is the electric field at the site of the radiator due to a 
x I 

->- 
single perturbing electon located at r . . 

In this manner we have incorporated the effects of the radiator 

charge into the dynamical treatment of the perturbing electrons. The 

classical path theories account for these effects by using hyperbolic 

trajectories in performing the perturber averages. 



APPENDIX B 

THE ALGEBRA OF TETRADIC OPERATORS 

The operators L, L Q , L^ R (oj) , K er (w) that we have encountered in 
evaluating J(ou,e) are special operators and are called tetradics. In a 
matrix representation defined on the system of radiator plus electrons, 
they have four pairs of indices. An operator of the usual type defined 
on this system will have two pairs of indices. For an example consider 
the Hamiltonian H : 


H Q " \ + H^ . (B-l) 

Matrix elements of H are given by 

<yo|H n |u l a , > = (E r + E 6 ) 6 ,6 , , (B-2) 

0' y a uu aa ■ ' 

where u,y' refer to the state of the radiator and a, a' refer to the 

electrons. The Liouville operator is a special tetradic defined by its 

action on an arbitrary operator f 

li = [H,f] . (B-3) 

From this definition, we can see that the operation of L on f will 
produce another operator with four indices. This means that products 
involving tetradic operators will have the following form: 

<ya| Lf|uV> = v6 J, 3 , [U uaill , 0?;v g vt g f vBv , B , , (B-4) 
<Ua| R Q (.) lJ|u»a»> = ve J, B , y „J v „ B „ [Bo(«)] vatV . B , ;vev , B . 



X[l] vf3v'3';y"a"v"g" f y"a"v"g" ' (B_5) 

Let us consider the following 

<ya| i^fly'a^ = <ya| [H QS f ] \ y f o*> 

124 



125 
= <ya| HgfJyV* - <pa|fH |y'a'> 

- (E r + E 6 ) <via|f |y'a'> 
y a ' ' 

- (E*, + E^,) <ya|f|y'a'> 

= (AE^, + AEj,) <yct|f|y'a'> . (B-6) 

We have obviously chosen the y,a states to be eigenstates of H . In 

order to introduce another notation, we now rederive the result of Equation 

(B-6). 

<ya| Lf |y'a'> = S [L 1 , , „ , , , , 
1 |M J ya;y'a';v3v'B' vBv'B' 



E {<ya|H_|vB>6 , ,6 , ,-5 6 <y V lH*| v' B '>} 
, IO , '0' y v a B yv aB '0' 



vBv 



* f l0 ,.. . • (B-7) 



VKV 



The expression within curly brakets defines the matrix representation for 
the Liouville operator. This equation may be simplified. 



<ya|L n f|y'a'> = E {(E r + E & ) 6 6 Q & , ,6 , fl , 
' „ fl „t(H M a yv aB y'v' a'B 



-<S <$ fl 5 , ,5 , Q , (E r , + E 6 ,)} f n , ol 
yv aB y v a y a T/ vBv'B 



E 6 6 „5 , ,6 ,„. {AE r , + AE e ,} f „ r „, 
, , yv aB y'v' a'B' py' aa' vBv'B' 



= {AE r , + AE e ,} f , , . (B-8) 

yy aa ya,y a 

We now consider the case where f = p g and g is an operator on 
radiator coordinates only. Using the above result we may compute the 
electron average of the product L~f . 






126 



<L O f V = E <ya|L P e g|y'a> 

a 



= I AE , <pap g|y'a> 
a 



- AE^, <y|g|y'> .. (B-9) 

We have used the fact that 

Tr {p }' = 1 . (B-10) 

e e 

We can recognize that Equation (B-9) may be also written in the form, 

<L S> yy' = <y l*- r g|u'>, (B-ll) 

or, in terms of operators, 

<L Q > = L r . (B-12) 

Proceeding along the same line we may also show 

<L 2 Q > = l\ , (B-13) 

and so on to higher orders. The goal of this procedure is to enable us 
to evaluate 

<R (w)> = <{w-L /R} _1 > . (B-14) 

If we expand the right hand side of this equation we obtain a series whose 
leading terms contain the expressions found in Equations (B-12) and (B-13). 
We may perform the indicated averages and resum the expansion to obtain 

<R Q (w)> - R°(uj) = {co-L r /h} _1 . (B-15) 

Employing the same type of expansion procedure, we may use Equations 
(B-7) and (B-8) to write down the matrix representation of R„(w) : 



.E r - ae 6 r 1 , 

V J yay 'a' ;vgv' g' ^ii 1 aa' u yv ag y'v'^a' 



[Rndo)],,,.,. ,_ Q „, ot ={u)-AE_, - AE_ , } &„J^ D S„ ,..,«_, , . (B-16) 



127 

This result allows us to prove two useful identities. As before, we 
set f = p g where g is an operator on the space of radiator states. 
We wish to calculate quantities such as 



<R (id) Lg > , = E <ua|R (to)Lp g|y'a> 
a 



avev'V u'W'S" ^W^vgvV [L Wb • ;y Wg" 



E E E {u>-AE r , - AE e } 6 5 6 , ,6 , 

a vBv'6' y'W'3" yp aa yv aB u v aS 



X tL] vBv , 6';y"a"v"3" %"a"v"&" (B_17) 

We may perform the sum over v3v'3' with the result 

<R n (w) Lg> , = E E {tu-AE r } 

° yy a y"a«V'3" yy 



X [L] yay'a;y'V'v"3" V'a'V'B" 

i r -1 

= E <ya {(jj-AE ,} Li ya> , 

yy 

a 

" 

= E <ya| R (id) L f |ya> , 

a 

= <R°(to) ^g> yu . • (B-18) 

In operator form we write this 

<R Q ((o)L> = R°(o))<L> . (B-19) 

Now let us consider 



X f y"a"v"S" ' 



128 



<LR Q (co) g> , = I <ya|LR (a>) f |y'a> 

a 



e ,-1 



z S 2 [L] , „ ,., {a)-AE r , - AE® ,} 



X W 6 vV'V3" f yVW ' (B " 20) 

We may perform the sum over u'W'g": 

<LR (m) g > , = S E [L] i . a ,., {w-AE r , - AE^ Q , } _1 
yy a vSv'S' yay a ^3v'B' vv ' 33' 



X f v3v'3' . (B21) 

Remembering now that f = p g and 

e 

W3' = P e V S W ' (B " 22) 

p 

we see that the delta function S , means that AE , will vanish. This 

pp 33 

gives 

<LR n ( w ) g> , = E E [L] , „ ,., {cj-AE r ,} _1 

uy a v3v'3' Pay'a;v3v'3 vv' 



X f v3v'3' 

= E <ua|LR (to) f |y a> > 
a 

= <LR°(a)) g > w , • (B-23) 

In operator form we write this 

<LR (u))> = <L> R°(co) . (B-24) 

It is convenient to summarize the identities which we have proven 
here. In terms of operators we have: 



<R Q (w)> = R (<a) ; 



129 
(B-15) 



<R (co) L > = R (oj) <L> 



<LR (co)> = <L> R (a)) 



(B-19) 
(B-24) 






APPENDIX C 

QUANTUM MECHANICAL PERTURBER AVERAGES 

In this appendix we carry out the electron averages of the operator 
expressions appearing in the line profile calculation of Section I, The 
basis set for the gas of electron perturbers is chosen to be a set of 
unsymmetrized products of single-particle functions. These single- 
particle states are the positive energy solutions of the Schrodinger 
equation for an electron moving in the field of a hydrogen-like ion of 

charge x = z-1. An expansion in spherical harmonics of these functions 

. 37,38 
is given by 

*k. " <X >V = " ~ZT3n (i)£ eiaU ' ki) Ov\m^V k i r) ' ((>1) 

l £m (2tt) 

where 

aU,k.) = arg r («, + 1 - in) ; n = - X- ; (C-2) 

i 

k. is a unit vector in the direction of k. ; J (k.r) is the Coulomb-Bessel 
function given by 

TT 

J £ (k.r) = e 2 r(S+2) (2k i r) ^ e"^ ^(i+l-inj 2£+2;2ik.r) , (C-3) 
where F signifies the confluent hypergeometric function. We find that 

J (k.r) reduces to the usual spherical Bessel function j (k.r) when x 

39 
vanishes. In fact we have 

Um ik = (2ir)~ 3/Z e ik i r ; (C-4) 

x-o k i 



i 



i> , $ dx = <5(k'-k) . (C-5) 

rC K. 



An n-particle state is represented by an unsymmetrized product: 

130 



131 

^ n; ($) = II <J» k (1 ± ) . (C-6) 

1=1 i 

The first problem we consider is the normalization factor for the 

electron perturber density operator. From Section I we have 

-1 -6H° . „, 



P e - c " e e , (C-7) 



where 

le 
e 



c = Tr {e 3H e} . (C-8) 



Now c is given by 

<k| e 
k 



c = E <k| e 6H e Ik > , 



■J 



Ai 



= [ \dx. E ** (x.)^ (x.) e B 2m ] N . ( C -9) 

J k. i i 

i 

The sum over k. resembles the Slater sum for an ion-electron system. 

The two-particle Slater sum for this system as given by Barker is 

2 2 
h k 

E 4>* (x)ijj (x)e _0E n + f <Wx)t|i, (x)e" e 2m 
/ s n r n r n k k v r k 

g(x) = — — ______ 



hV • (c " 10) 

E <jjg (x) ip (x) e B 2m 

k 

The sum over n gives the contribution of the bound states. The sum over 
k is the term appearing in the square brackets above. The denominator 
gives a normalization factor where tJj is a plane wave state. By neglecting 
bound states in our trace operation, we make the following approximation 



= [ jdx. g(x. 



2 2 

} J *6 *0 e -2m~ ] 
k. 
i 



= [ H 1 N , (C-ll) 






2,2 

"* , (N) ,->, (N)-\ - A— — - 

dx E # uv xv, U)e 'H^T 

, k k 
k 






132 
where V is the volume of the system and A is the thermal wavelength of 
the electrons. Hence we now have 

o - I ' i i» -BH° 

P e ~ ' ■ V J e e • (C-12) 

The neglect of bound states implies that we neglect the probability that 
a perturbing electron will become bound on the radiator to form a 
doubly-excited state. This approximation is discussed in References 
9 and 23- There it is pointed out that the neglect of bound states has 
essentially the effect of introducing a close collision cutoff into the 
calculation. 

The next result we wish to obtain is the proof of an assertion 

made in Section I that the electron average of L vanishes. We will 

er 

make use of the results given above. From the definition of L we may 

er 

write 

<L > f = <V > f - f <V > , (C-13) 

er er er ' KU ^ JJ 

which tells us that we are interested in calculating <V > or, 

er 

<V > , = Tr {V , p } (C-U) 

er uy' e uu H e ' ^ ± ^ ) 

Since <V er > is an operator on the Hilbert space of the radiator, the 

indices u,u' refer to radiator quantum numbers. The trace may be written 



[^A^e-^r fdxV (N) (x)V ,^ N > 
J ^ k w k 



^erV' = hr r \ dk " e " 2m |dxV UV (2)V M ,^ W Cx) . (C-15) 



rh<2 V er iS wri - tten in terms of spherical tensor operators as follows: 
where the dipole approximation for the radiator-perturber interaction has 






133 
been made. The sum over i includes all of the electron perturbers. If 
we then insert this above we obtain 

3 r t 2 ! 2 

q „ 2 D (q) 



<V > , = E (-l) H e R VH , [— -] Idk e 2m 
er up . pu 



,, [— 1 Jdk ■ 



\ 



r (-q) 

dx ijj k (x) — ^ ijj (x) . (C-17) 

r . 
i 



The 2N-dimensional integral on the right side of this expression may be 

broken down into two factors. One factor consists of an integral over 

k. and x. - essentially a single-particle matrix element involving the 

i-th perturber. The other factor may be recognized as just the right 

side of Equation (C-9) with an exponent of N-l instead of N. The result 

of this factorization is that we are able to cancel N-l of the factors 

inside the square brackets. 

3 r k2,2 



nX T 2 

<V > , = e 

er py N 



S R k i 

where F. is the single-particle matrix element of the i-th perturber 

part of the interaction V 

er 

f r (q) 

F i q) <W - K v <*i> -V \A } • <c ~ 19) 

J 1 r. 2 



Now using the definition of r 

l 



l 
(q) 41 



■ I? 



r (q) ■ Y ',;::, 



and the spherical harmonic expansion in Equation (C-l) , we may perform 
the integration with the following result for F. : 

4 q) (k r k 2 ) = \ i % c-i) £ i e - ia{£ i'V(i) £ 2 e ^a 2 ' k 2) 

Vl £ 2 m 2 






134 
X \ m] _ ( V Y , 2 m 2 Ck 2 )(-l) m l [(2 4l + i)(2t 2 + 1)] 1/2 

£ 1 - 1 £ 2 £ l 1 £ 2 

X ( } U q m > T £ •£ ' • «>21) 

1 H 2 1' 2 

In the expression, k. and k are unit vectors in the direction of k and 
k , respectively. 

CO 

\;* 2 - ] dr r ' \ (k i r) 7 % (k 2 r) . < c - 22 > 

o 
is the radial integral. Inserting the appropriate expression back into 
Equation (C-18) we have: 

2 A 3 f R 2 k 2 

q J £ i ,m i £ 2 m 2 

x (i) 2 e - ( V k) Y^Ck) Y £ ^ (k)(-l) l [(2Al + 1)(2, 2 + 1)] 1/2 

-m -q m' l,\l 

We now point out that since p is independent of angles (isotropic) we 
are able to perform the angular-k integration. This gives delta functions 
in &-.J&0 and m ,m — causing the leading phase factors to cancel to unity. 
We obtain 

CO 

3 2 2 

2n V t \ C o „Rk 

<V > , = 1 E (-l) q R (q I dk k 2 e~ 3 "2m" E (-l) m 

« q My« J to 

o 

x (24 +1) l ; -m -q m'' T . (C-24) 

,1 1 L 
However, vanishes because of angular momentum rules. Therefore 

each term vanishes and we have the result 






135 

<V er > = ° and <L er > = ° ■ (C_25 > 

Before continuing, we point out two facts which caused <L > to 

er 

vanish. First the fact that the perturber distribution is isotropic 
allowed us to perform the angular-k integration. This, along with the 
dipole approximation for the radiator-perturber interaction, caused 
<L > to vanish from angular momentum considerations. We note in passing 
that a more careful treatment of <L > not making the dipole approximation 
yields a small but nonvanishing contribution that may be related to the 
plasma polarization shift. 

We now turn to the evaluation of an expression appearing in the 
width and shift operator H (u)) . The expression we wish to calculate is 
given by: 



I M ,1 , = <V „ V „ ,(t)> 
yy";y"y' yy y .'V 



. t ^i , [Md^.-«4£.- 1 4<4-$ 



•<Mk 1 |v|n H k 2 > <y"k 2 |v|y'k 1 > 

3 2 2 

nA T f *N f "^w p H k -r*JL n 2 i 2 ^ * * 

= [~~] E \dk^ dk e P 2m e lfc 2m U 2 " Va. (k. ,k,) „ 

im JJ 112 \iU 



' A m (k 2' k l>y"y' ■ (C ~ 26) 

where A. is the many-body matrix element of the interaction between the 
radiator and the i-th electron perturber. 



A i (k 1 ,k 2 ) = e 2 E(-l) q <y|R (q) |y">[ II 5 . (k.^) ] F< q) (k^ kj . (C-27) 

Here, F. is just the same matrix element defined in Equation (C-21) . 
First consider the N terms in Equation (C-26) where i=m. Denote this 



136 



contribution by I : 



3 r- /- JL 2 



V';yV N[ ~ ] e J dk i J dk 2 e ^T e 2m 2 x 



■ i (-i^+q' x <y|R (q) |u"><y"|R (q,) |y'> 
q>q' 



-y -y 



■[ fl 6 (k r k 2 )] 2 f[ q) (k ,k ) F^ q ' ) (k 2> k 1 ) . (C-28) 

The squared delta functions allow us to perform N-l k -integrations and 
N-l k„-integrations with the result that N-l of the leading factors 

—r— - are cancelled. The result is 

N 

V;m'V =nA x e E , (-l) q+q ' <y|R (q) |y"><y"|R (q,) |y'> 
q,q 



Kl 



2 2 
"*■ _ S H k 1 _ -.JL/-1 2 i 2 ^ ( \ -*■ ■*■ t iv •* •* 
dk 2 e 3 ~2~ e xt 2^ U 2 V f| ^(k^k^FJ: q '(k^) .(C-29) 



The many-body problem has been compressed down to a calculation involving 
only single-particle functions. We now may insert the definitions of F. 
from the Equation (C-21) . 

Once again we may take advantage of the isotropy of the perturber 
distribution. That is, we may again perform the angular-k integrations. 
The result is that the phase factors cancel to unity and we are able to 
perform several sums: 

1° .... =^nA^ e 4 Z (-l) q+q ' <y|R (q) |p"><y"|R (q,) |y'> 
7r q,q 



00 CO 

* 1 dkl I 



2 2 
„ „ h k- Ti , 2 2 m +m 

dk„ k, k„ e — ^ — e 2m 2 1 E £ (.-1) 

2 1 2 2m „ . 

1* 1 2™2 

o 






137 

l l 1 l 2 2 °l 1 l 2 l 2 1 £ 1 
x [(2l ± + 1)(2£ 2 + 1)] ( } ( - mi -q m^ ( -m 2 -q ' m^ T £ . £ T £ . £ 

(C-30) 
There is some cancellation of the phase factors because of a rule for the 

1 • IT 41 

3-j symbols : 

-m.-q + m 2 = . (C-31) 

Thus , 

(-l) q (-l) m l +m 2 = 1 . (0-32) 

Now we are able to perform the sum over m ,m with the aid of an identity- 
given in Edmonds : 

. • • « * > ^ 

( 3 1 2 2 J 3 )( J 1 3 2 J 3 ) _ ± 

S m m m„ m. m„ m' = (2j „ + 1) S . . , 6 , ,_ „„. 

m r m 2 12 3 12 3 J 3 ^ m^ . (C-33) 

The result of performing this sum is 

4neV 

1° „ „ , =-^ E ( q' <ylR ( -^ ) |y"><u"|R (q,) |y'> 

3it q 

2 2 

03 oo H k 

- 1 .J! ,, 2 ,2, 
-it-r- (k.-k. ) 



Hi 



\ dk, \ dk 2 e P 2m e 2m '2 V f (k ,k 2 ) ,(0-34) 



o o 

where 

l l 1 £ 2 2 

f(k ,k ) = kV E (2£ + 1)(2£ + 1) ( } T. .. T .(C-35) 

i z l z £ ^ i z ^,^2 * 2>Jtl 

The sum over q' is recognized as the spherical tensor notation for the 
vector dot product. One of the sums in Equation (C-35) may be performed 
and the 3-j symbol evaluated explicitly (from Edmonds ) with the 

result 






138 

f(k l'V " k l k 2 l { ^ + l > T , ; A 1 + *!,. A x > • (0-36) 

We now need to go back and consider the terms in Equation (C-26) 
where i/m. We denote this by I : 

i u - e 4 ^ r n ^i N f d ,> f *> "4^ -itf (k 2 -k 2 ) 

Wmv e 2 [ ^T ] J dk lJ dk 2 e 2m e 2ra 2 1 • 



X n E n ,(-i) q+q, <y|R (q) |y ,, ><y"|R (q ' ) |y"> n 5 .(k -Uf!^ (I ,kj 

->• •> ■*■-$- 

x n 6 (k -k ) F^" q ' ) (k„,k 1 ) . (C-37) 

_/_ n l i m 2 1 

nfm 

In this equation we have N-2 squared delta functions. Thus we may perform 
all of the k - and k -integrations except those corresponding to particles 



i and m. 



V'jyV " [ 1T ] J dk l \ dk 2 e 2m e 2m 2 1 



x E (-l) q+q ' <y|R (q) |u"><y"|R (q ' ) |y"> 6 (k -k.) 6. (k -kj 
q,q' m 1 2 i 1 2 

x F^d^.kj) F^" q ^kj,^) . (C-38) 

The remaining delta functions allow us to perform the k„-integrations . 



Then we obtain 



2 2 

, N(N-1) nA^ f ■+ H k l 

^■ ; yv = e 2 ^ 2 J dk i e ~ 6 ^ * , ^> q+q <p|* (q V> 



x <y"|R (q } |y'> Fj'^Ck^k^ F „" q ' > (fc 1 »k 1 ) - (C-39) 

Comparing this result with Equation (C-18) we can identify the terms 
above as 






139 

u 1 
I h, ti » = -r <V > „ <V > „ , = . (C-40) 

W ;y v 2 er yp er ]}"\i' v ' 

Therefore the neglect of correlations in the density operator leads to 
the result that the i^m contribution to I must vanish. The final result 
we have obtained is 



2 2 
4ne 4 A 3 f ^ * k l 

<V W " Vy' (t)>= -^2- R py"'Vy' \ dk l ) dk 2 e 2m 



. H_ ,,2 2. 
x e XC 2m U 2 V f (k ,k ) , (C-41) 



where 



f(k r k 2 ) « k^k 2 2 E {(£ + 1) T £; / + x + £T &;£ 2 _ x } . (C-42) 

(24) 
We conclude here by identifying f (k.. ,k.) as 



f(k l5 k 2 ) = k 2 k 2 2/3 g ff (k 1 ,k 2 ) , (C-43) 

where g (k ,k ) is the free-free Gaunt factor. 2 ' 9 ' 23 ' 42 ' 43 



APPENDIX D 

THE PARABOLIC REPRESENTATION 

Schrodinger 's equation for hydrogen-like atoms is separable in 
parabolic coordinates. According to Bethe and Salpeter - "This 
alternative is connected with the degeneracy of the eigenvalues belonging 
to like principal and different orbital quantum numbers. " The connection 
between spherical coordinates and the parabolic coordinates g, n, and <J> 
may be expressed by 

x = E,r\ cosij) E, = r + z 

y = 5n sine)) n = r - z 

1 —1 v 

z ■ ■=■ (£ - n) (j) = tan x 

r = ~ (| + n) (D-l) 

The details of the separation and solution of the Schrodinger equation in 
parabolic coordinates are given in some detail in Reference (11) . The 
normalized eigenfunction is given by 

±im<j, 1/2 1/2 m + 3/2 
e (n ! ) (n !) e 

n,n„m /it n 



'i "o"' 3/2 3/2 

1 2 [Cn x + m)!] J/Z [(n 2 + m)! J/Z 

- 7re(£ + i"l)/>. N l/2m _ m . _.„ m , . ,_ _,. 
; (?n) L ni+ m(^> L n 2+ m (erl) . (D ' 2) 

where £= and the L's are the associated Laguerre polynomials. 

na o 

n = n + n + |m| +1 } (D-3) 

where n is the usual principal quantum number. For fixed m, n (or n ) 
runs from to n-|m|-l. Also m runs from to n-1 . It is convenient to 



140 






141 
define a quantum number q: 

q = n l " n 2 ' (D " 4 ) 

The number q is called "the electric quantum number". Using these 
relations and the allowed values of n (and n ) and m, we can generate 
the quantum numbers n,q,m which specify the hydrogenic states in the 
parabolic representation. 

The Stark effect for hydrogen-like atoms is especially amenable to 
calculation within the parabolic representation. The interaction to 
consider is (see Appendix A) : 

V. nt = eze . ( D _5) 

The electric field e defines the z-direction of the atomic system. A 
first-order perturbation treatment of this interaction yields the energy 
level shifts of a hydrogen-like atom for the linear Stark effect. When 
this calculation is carried out the level shifts are given by 

3 a e 
AE nqm = 2 T nqE ' (D " 6) 

Note that the energy levels are still degenerate with respect to the 

quantum number m. This degeneracy is a consequence of the rotational 

symmetry of the perturbed Hamiltonian about the z-axis and remains in 

all orders of perturbation theory. 






APPENDIX E 

CALCULATION OF RADIATOR DIPOLE MATRIX ELEMENTS 

In this appendix we wish to derive a computational form for the 

radiator dipole moment matrix elements in Equation (1-94) . The expression 

we are interested in is 

-$■ -»■ 
D li , - <i|d|l>.<i|d|i'> , (E-l) 

where | 1> represents the ground state of the hydrogen-like radiator. 
Employing the parabolic representation introduced in Appendix D for the 
hydrogenic states i and i' we have 

D , = <nqm|d|l00>-<100|d|nq'm'> , (E-2) 

where n is the principal quantum number for the upper state of the 
transition and q andm.ar^the parabolic quantum numbers. With the notation 
of Edmonds we may expand the vector dot product into a product of 
tensor operators. 

D , - I (-l) k <nqm|d_ |l00><100|d |nq'm'> , (E-3) 

n k k k 

k i 
= E E E (-1) <nqm|n£ m ><n£ m |d |100> 
k «, 1 m 1 £ 2 m 2 L L Ilk 

x <100|d |n& m ><nl m |nq'm'> . (E-4) 

The state | 100> is equivalent in the two representations |nqm> and |n&-m>. 
Now we must evaluate the matrix elements of the spherical tensor operators 
d between spherical states. From Edmonds, 

d ±k - (f ) 1/2 er Y 1)±k (fl) . (E-5) 

142 



143 
Inserting this and performing the angular integrals of the spherical 
matrix elements we obtain: 



k 
D. .. = I £ X (-1) <nqm ni^m ><n£ m lnq'm'> 
11 . „ „ ' 11 2 2 1 

k ^ l 2 m 2 



X ( - 1} K2£ 1+ 1)C2£ 2+ 1)] (^ _, )( Q \ )( k m 2)(° J Q 2 ) 



x <n£ | er I 10><10 I er I n£ > . (E-6) 

The last two factors here are radial matrix elements of er. In obtaining 
this equation we have made use of the following identity 



\ 



da \* ± ™ %m 2 <°> Y , 3 m 3 < D ? 



(2£+l)(2£ 9 +l)(2£„+l) . A l 2 \ A % 2 \ 

= [ t-^ 1— ] 1/Z ^0 O'V m, m„ ; (E-7) 

4tt 1 2 3 ' ■ 



The 3-j symbols vanish unless the following angular momentum rules are 
met: 

(i) The triangle identity, 

K-JU i^i^ + ilj ; (E-8) 

(ii) m + m 2 + m 3 = ; (E-9) 

(iii) if m = m = m = , then 

I + I + I = 2n (n = 0,1,2,...) , (E-10) 

The important symmetry properties of the 3-j symbols are given by 
Edmonds : 






144 
(iv) An even permutation of the columns leaves the numerical value 
of the 3-j symbol unchanged: 

h h h m j 2 h h m h h h 
m l m 2 m 3 m 2 m 3 m i m 3 m l m 2 

(v) An odd permutation of the columns is equivalent to multi- 
plication by (-1)^1 ^2 ^3: 



( 1 2 3 ) = (-1) j l + V j 3 ( 2 1 h . (E-12) 

m 1 m 2 m 3 m 2 ^ m 3 

(vi) Also, 

i 1 3l 33 ) = (-l) j l +j 2 +j 3 ( X 2 3 ) . (E-13) 

m m m -m -m^ -m 

Now we note that, in Equation (E-6) ,£=£,_ = 1 because of rules 
(i) and (iii) . Also rule (ii) requires 

(-l) k+m l = 1 . (E-14) 

Thus the angular momentum rules satisfied by the 3-j symbols have 
allowed us to simplify Equation (E-6) considerably: 



D..i = 1 I <nqm nlm, ><nlm Inq m'> 
ii , ' 1 2' 

k mm 

x ( 1 \ °)(° I X ) |<nl|er|l0>| 2 . (E-15) 

-m -k k m„ 

41 
Some cancellation has occurred due to the following identity: 

( J j ) m ( _ 1} i-n. (2j + ir l/2 (E _ 16) 

(J m -m 

Now rule (ii) allows us to perform the sums over m- and m . 



D... ■ L <nqm nl-k><nl-k nq m > 
k 



145 

,1 1 W 1 1 N | , , 1 2 
x <v -v n>(n t -0 |<nl|er|lO>| . (E-17) 



^k -k 0^0 k -k 
Using rule (iv) we obtain 



D iit = £ <nqm|nl-k><nl-k|nq*m'> (J * °) 2 | <nl| er | 10> | 2 . (E-18) 
When this 3-j symbol is evaluated using the above identity we obtain 

D.., = Z <nqm|nl-kxnl-k|nq'ra'> -| <nl| er | 10> | . (E-19) 

A convenient form for <n£m'|nqm> is given by Vidal, Cooper, and Smith. 

n-1 n-1 

<n£m'|nqm>=6 , (-l) 1/Z(1 ^" n) [2£+l] 1/2 ( 2 J ) . (E-20) 

' mm' m-q m+q _ ' 

2 2 m 

Inserting this into Equation (E-19) we obtain 

n-1 n-1 .. 
D... - E {« (_Dl/2(l-k-q-n) j- 2 2 

n . m,-k * v m-q m+q ' 

~2 ~2 _m 



X{6 km'^ 1/2(1+m '" q, " n) I^C) . "I, \)> 
-k,m 'I m-q m+q -m 



1 9 

x — | <nl | er 1 10> | ( (E-21) 

C + » \ n ~^ n-1 , n-1 n-1 

= 6 , (-l) 1 ~ n+m 2 1 ( 2 2 )( 2 2 ) 

mm ' m-q m+q -m m-q' m+q ' -m 

2 2 2 2 

x |<nl|er|l0>| 2 . (E-22) 

Equation (E-22) is now in a form convenient for computation. The 3-j 

symbols are computed using a subroutine given in Reference (45). 






APPENDIX F 



THE MANY-PARTICLE FUNCTION r(Aoj) 



The definition of the many-particle function r(Aw) is obtained 
by combining Equation (1-76) with the result for <V V(t)> given in 
Equation (C-41) . Upon making this step we find: 



/ o 2 °° °° 2 2 

4 ™ 4 f r r K k i -u £-(k 2 - a 

r(4 ., . _i f_J] ) dt J dk j dk ,u«t.-( -^ e 2/2 k i> f 

3tt 12 



(F-l) 
Here n is the density of the electron perturbers in the plamsa; A 
is the thermal wavelength of the electrons; = (kT)" 1 for the electrons. 
The function £(1^, kp was defined in Appendix C. It is related to the 
free-free Gaunt factor 42 ' 43 by the following: 23 



f(k x , k 2 ) = k 1 k 2 — g ff (k v k 2 ) . (F-2) 



We insert a theta function into Equation (F-l) in order to extend the 
lower limit on the t-integration to -». 



OO 00 CO 

4ne A 

r(Aaj) = -i [ ^L] 

3tt 
2 2 
. Ek l 



00 00 CO 

\ dt\ dt ± \ dk 2 G(t)e 1Awt e- it: fe (k 2 " k l> 



• e 2m fCkp k 2 ) , ( F _ 3 ) 

i ^V^ J ^ \ dk \ dk„ ^r]dz V^ e 1 ' 



iAut 
- 

z+iri 



2 2 

B 2 2 ^ k 1 

-it £- (fcf - kf) -f 



2m 2 1 e 2m f(k r k 2 ) . (F-4) 

146 






147 
We have inserted the integral definition of the theta function. 16 
We define' n to be the positive infinitesimal. We may now perform 
the t-integration in Equation (F-4) . 



OS CO co 2 2 

r(A.) = l^l£l J dz ^ 5 dk i 5 ^ e "^ fCk l'V 



* 6 {Aa, ~ Z "'2m~ (V k l )} • ( F ~ 5 ) 

Let us now define a new function of Aw: 

°° °° 2 2 

. 4.3 f r ? k i 



5 dk i 5 



G(Au>) = -tt[ T ] \ d^ \ dk 2 e p 2m f(k ,k ) 

3tt 2 



' 6 {Aal " to (k 2" k l )} ' (F_6) 



With this definition, Equation (F-5) becomes 



CO 

5 



r(Aco) = - - \ dz —V- G(Aoj-z) . (F-7) 

At this point we apply the following identity 

-^ -> P \ ~ 1tt5(z) . (F-8) 

We then obtain 



r(Au) 



00 oo 

_I P I dz G(Au-z) + . [ dz G(Aw _ z)6(z) ^ (p _ 9) 

—CO —CO 

CO 

» p J dz S£ + iG(AW) • (F - 10 > 



This equation allows us to identify G(Aoi) as the imaginary part of the 
function r(Aio) : 



148 

r im (Aa>) = G(Aoj) ; (F-ll) 



CO 

■i'S 



r T (z) 

r_ (Aw) - - P \ dz m . . (F-12) 

Re n i z-Aw 



We now must develop a computational form for the function T (Aw) , 



where 



00 oo 2 2 

4ne \ [ \ „ 



5 dk i J 



r im (A„) - -„ [— ji] \ d kl dk 2 . 2m £(k ,kj) 

JIT *' 

o 



•6 {Aco- ~ (k^-k*)} . (F-13) 

We now insert the definition of Equation (F-2) and make the following 
change of variables: 

k i - 4- h ■ h - zr K 2 • < F - 14) 

n h 

We obtain 

4 3 
4ne \_ 2 . „ 3 
n tk \ r Tt r me , 4 r 2n , 
r T (Aw) = -it [ =-] [— ] [— rl 



Iiri ' L _ 2 J \2 J L 4 J 2/3 
3tt h me 



CO oo 2 , 

f f _K 1 /6 R_ 2 2 

\ d^ I dK 2 e "KjK g ff (K 1 ,K 2 )6{An-K 2 + Kj} , (F-15) 



where 



3 2 

Afi = ~- Ao) ; 6 R = -25- K B T ; (F-16) 

me me 

are the frequency and temperature in Rydberg units. We consider first 

the case where Au)>0. We employ an identity from Messiah to simplify 

the delta function with the result 



149 



4ne X 2 . ,_3 



Im l . 2 Jl 2 J L 4 J 2/3 

3tt H me 



00 CO 

J dK l J dK 2 e ~ Kl/6R K i g ff ( K r K 2 > « <X X -K 2 ) , (F-17) 



x \ dK 



where 



2 | ', 2 
X 1 = |Afl| + K ± . (F-18) 

The integral over K may now be performed. After some algebra we obtain 

CO 

4 r 2 

r M<i)>n\ - r 2ne ■, r 8Trm -|l/2 2-n \ -k /Q„ 

l jJ*" 0) - - [-y-H-j^-] 7 y 7 - j d k;L e l'^K^OL^X^CF-lQ) 

o 

The one-dimensional integral in this equation may be evaluated rapidly and 
accurately using a Gauss-Laguerre quadrature formula. 

Proceeding in a similar manner and using the fact that g f (k ,k ) = 
g ffr (k ,k ) we obtain 



r im (Ato<0) = e l An l /0 R r im (Aoj>0) . (F-20) 



After computing T (Aw) in this manner we may obtain T (Ao)) by 
lm Re 

numerically evaluating the Hilbert transform of Equation (F-12) . 



APPENDIX G 

A COMPUTATIONAL FORM FOR THE ATOMIC FACTOR 

We indicated previously that the electron broadening operator 
for our case will factor into a frequency-dependent many-body part 
times an atomic factor. The many-body function r(Au) was discussed 
in Appendix F. As a prelude to the development of a computational for 
the atomic factor we now consider 

<nlm A tt2'm'> = „ -^mlRln^m > • <n£. m J R ! n£ 'm ' > . (G-l) 
g m ''11 ll 11 

This expression gives the form of the atomic factor in the spherical 
representation. That there is no internal sum over n here is a 
consequence of the no-quenching approximation. Using Equations (E-3) , 
(E-5) , and (E-7) we obtain 

<n£m|A|n£'m'> = £ E (-l) k <n£|r|n£ ><n£ |r|n£'> 
k £,m 

A/X/Aj-- JO JO JO _ 

x (-l) m + m l [(21 + 1)(2£ 1 + 1)] 1/2 ( )( -m k m } 

I IV % 1 £' 
x [(22 + 1)(22' + 1)] 1/2 ( 0^-B^ -k m*- 1 . (G-2) 

Rule (ii) (Appendix E) for 3-j symbols causes the phase factors to cancel 
to unity. If we also use rules (iii) , (iv) , and (vi) to rearrange this 
expression, we obtain: 



<n£m|A|n2'm'> « Z E <n£|r|n£ ><n£ |r|n£'> (2£ + 1) 
k 2,111 

tli £ 1 £, £' 1 2. £' 12. 

1/2 ( )( )( )( X ) 
x [(22 + 1)(2£' + 1)] /Z S 0, -m k m 1 V -m' k m ± J _ (G-3) 

150 



151 



41 
The following identity allows us to perform the sum over k,m 1 : 



( 3 1 J 2 3 3 ( 3 l J 2 J 3 } 

E m m m ; V m ml. = (2j + 1) 6. ., 6 
m im2 1 2 3 1 2 3 3 J 3 J 3 m 3 m 3 



(G-4) 



After performing this sum we obtain a much simpler expression. 

tl£ 12 

<n£m|A|n£'m'> = 6 5 , E |<n£|r|n£ >| 2 (21. +1) ( ? . (G-5) 

&*» nun n J_ J_ 

*i 

Since the 3-j symbol must satisfy rules (i) and (iii) , the sum over £ 
has only two terms. Performing this sum we obtain 

2 A 1 £-1 2 

<n£m|A|n£'m'> = 6.., 6 , { I <n£| r |n£-l> I (2£-l) " > 
££ mm ' ' ' ' 



2 ,£1 £+1.2 

+ |<n£|r|n£ + 1> I (2£ + 3) { 6 } 



(G-6) 



The 3-j symbols here are a special case evaluated in Edmonds. After 
substituting in the values and performing some cancellation we have 



<n£m|A|n£'m'> = 6 ££l 5^, { | <n£| r | n£-l> | 2 j^- + | <n£ | r |n£+l> | 2 ^.(G-7) 



The radial matrix elements are given by Condon and Shortley: 



47 



CO 

s 



3 a n 2 2 1/2 
dr r R(n£) R(n £-1) =4 -**- [n - £ ] ' 

z z 



(G-8) 



where a is the Bohr radius and z is the nuclear charge. When this 
expression is used in Equation (G-7) we find 

q 2 2 

<n£m|A|n£'m , > = 6 flflt 6 , — 2— -±— {[ n 2 -£ 2 ]£ 
11 ££ mm , 2 2£+l 

4z 



+ [n 2 -(£+l) 2 ] (£+1)} 



(G-9) 



After collecting terms we arrive at a convenient form. 

Q 2 2 

n 11 2 2 

<n£m|A]n£ I m'> = 6 n „ , 6 , — ^— [n -£ -£-1] 
££ mm . I 



(G-10) 



152 
Our intent here is to obtain a computational form for the atomic 
part of the electron broadening operator. Now we will show that Equation 
(G-10) is a valuable step in the development of a computational form for 
the atomic factor in the parabolic representation. We wish to compute 

<nqm|A|nq'm'> = I <nqm|R|nq m > -< n q m |r| nq'm'> . (G-ll) 
q l m i 

Now the sum over q_m , that is, > 

Inq.m , ><nq ,m, I 
q^ ' U 1 H l l 1 

is just a projector onto the subspace of a given n. Therefore 



qm l n( l 1 m 1 ><n< l 1 m 1 l = z [ni^xnA a^J ( (G12) 

11 £ m 



<nqm|A|nq'm'> = E <nqm|R| n£ m >• <n£ m | R|nq 'm' > (G-13) 

Vi 



I I 1 <nqm|n£ m><n£ m |R|n£ m >-< n £ m |R|n£ m > 

£ A £ 2 m 2 £ 3 m 3 U 2 2 



x <n£ 2 m 2 |nq'm> t (G-14) 

Here we may recognize that the two middle factors are just the spherical 

representation of the atomic factor. We now insert the result appearing 
in Equation (G-10) . 



<nqm|A|nq'm'> = I E <nqm|n£ m > 

£ 2 m 2 £ 3 m 3 



>' <n£ 2 m 2 |A|n£ m ><n£ m |nq'm'> f (G-15) 



153 

Q 2 2 

= y o 1 * 7 7 

% m <nqm|n£ 1 m 1 > — [ n -1^-4-1] <nl^n |nq 'm'> . (G-16) 

11 4z 

We again need to make use of Equation (E-20) . We then obtain 

<nqm|A|nq'm'> = I {6 (-1) l/2U+m-q-n) [2£ + 1] -l/2 

Vl X 



n-1 n-1 . _ 2 2 
£., 9a„n 



x ( 2 J ^-^-[n 2 -^-!] 
m-q m+q ,2 11 

— n — n _ m 4z 



n-1 n-1 



x 



{ ( 1/2(1^-, *-n) + 1/2 2 2 1 

m,m 1 m-,-q m+q' -m , 



9^n 2 1+m _ n .(q+q') 

4z £ 



n-1 n-1 n-1 n-1 

2 2 2 ? 

m-q m+q -m y v m-q' m+q' -m y [n -£ -£-1] . (G-18) 

2 2 2 2 

This is the final form for the atomic part of the electron broadening 
operator. 



APPENDIX H 
NUMERICAL PROCEDURES 

Introduction 

The first two sections have developed the theory needed in computing 
Stark broadened spectral line profiles. The numerical procedure for 
producing these profiles involves four basic steps. This appendix will 
provide a discussion of the four programs needed, as well as the 
computational techniques involved. The first three programs carry out 
the production of the electric microfield probability distribution 
function P(e). The fourth program computes the Stark broadened profile 
for a given set of plasma parameters. Also, this final program generates 
plots of the Stark profiles and the Doppler-corrected profiles. 

In addition to discussing the numerical techniques employed, we 
discuss sources of error and the general reliability of the programs in 
their present form. 

The Alpha-Search Program 

In Section II we indicated that a is a variable effective range 
parameter. Essentially, a determines how much of the calculation will 
be treated by the collective coordinate method. It enters the calculation 
in the following manner: 



N l N 2 
V - V„ + I W.„ + I W n , (H-l) 

. . i0 mO 
j=l J m=l 

where V is the total potential energy of the plasma and W.. is given by 

154 



155 

W. n = XZn 7~ e jO . (H-2) 

J JO 

We recall that these W's are the short-range central interactions which 

are treated by cluster-expansion techniques. The term V defined by 

these two equations contains long-range central interations as well as 

all the noncentral interactions- This term is treated by the collective 

coordinate method. 

By inspecting these two equations and comparing them with Equation 
II-9, we can see that as a approaches unity, the contribution of central 
interactions to V vanishes. In this limit, then, V contains contributions 
only from the noncentral interactions. In this sense, as we stated 
previously, the parameter a "measures" the relative importance of the 
short-range interactions and the collective coordinate contribution. 

In the calculation of P(e), two basic approximations have been made. 
First, we have terminated the cluster expansion at second order. Second, 
we have neglected correction terms in the Jacobian of the transformation 
from spatial coordinates to collective coordinates. As is well discussed 
in the literature, > > t ^ e a pp ro p r i a te choice of the value of 
a should make negligible the error in P(e) due to each of these two 
approximations. If this is the case, we should be able to locate a range 
of a values for which the final computed P(e) is stationary. Indeed, 
this is the case. The a-search program attempts to locate an "a-plateau" 
region over which T(L) is stationary. In all cases considered here, 
there is an obvious a-plateau region. 

The a-search program currently requires 256 k bytes of memory on 
the IBM 370-165 at the Northeast Regional Data Center at the University 
of Florida. The program computes T(L) in the first approximation at 
several L values (currently four values) for each a value. A typical 



Table H-l. The electric microfield distribution 

function P(e) is tabulated for different 
values of a, the effective range parameter, 
a = 0.8. The radiator is hydrogenlc argon 
(X= +17), perturbed by ions of charge +1. 



TABLE. H-l. 



157 









P(E) 






E 


a=0.95 


a =1.05 


a=1.15 


0-1.25 


a-1.35 


0.1 


0.235 


0.237 


0.240 


0.239 


0.232 


0.2 


0.758 


0.763 


0.759 


0.767 


0.751 


0.3 


1.223 


1.227 


1.232 


1.230 


1.215 


0.4 


1.430 


1.431 


1.431 


1.429 


1.422 


0.5 


1.387 


1.384 


1.382 


1.380 


1.381 


0o6 


1.196 


1.193 


1.189 


1.188 


1.194 


0.7 


0.959 


0.955 


0.952 


0.952 


0.959 


0.8 


0.734 


0.731 


0.729 


0.729 


0.736 


0.9 


0.546 


0.544 


0.543 


0.544 


0.549 


1.0 


0.400 


0.399 


0.398 


0.399 


0.404 


1.1 


0.291 


0.290 


0.291 


0.292 


0.295 


1.2 


0.212 


0.212 


0.212 


0.213 


0.215 


1.3 


0.154 


0.155 


0.155 


0.156 


0.158 


1.4 


0.113 


0.113 


0.114 


0.114 


0.116 


1.5 


0.084 


0.084 


0.084 


0.085 


0.086 


1.6 


0.062 


0.062 


0.063 


0.063 


0.064 


1.7 


0.047 


0.047 


0.047 


0.048 


0.048 


1.8 


0.035 


0.036 


0.036 


0.036 


0.037 


1.9 


0.027 


0.027 


0.028 


0.028 


0.028 


2.0 


0.021 


0.021 


0.021 


0.021 


0.022 



158 
run for 20 a values requires vL.2 minutes of computer time. 

Since we employ the first approximation to T(L) , the major part of 
the program is directed at computing the terms I 1Q (L) and I (L) . These 
one-dimensional integrations are performed by subdividing the range of 
integration and using a 32-point Gauss-Legendre quadrature formula in 
each sub-interval. This procedure has been tested for numerical accuracy; 
results indicate that we have obtained at least six significant figures 
of numerical precision for I, Q (L) and Ig-rCL). Thus the numerical 
precision present in the a-search program is comfortably redundant. 

Presently, the choice of a requires a decision by the programmer. 
A future goal is to develop the program to allow the machine to make this 
choice. After an inspection of the output for T(L) , it is a straight- 
forward task to locate a range of a values for which T(L) is approximately 
stationary. In certain cases (especially a = .6,. 8), the a-plateau 
appears to be quite narrow. However, as is indicated in Table H~l, the 
final curve for P(e) is much more stable with respect to a variations than 
might be expected. In addition, the effect on P(e) due to a variations 
offers one type of error estimate for the final P(e) curves. From this 
point of view, we may expect that errors in P(e) will not exceed a few per 
cent. 

The P(e) Production Program 

The selected a value is fed as input into the P(e) production 
program. This program requires 384 k bytes of storage and may require 
from ^4.0 to 7.5 minutes of computer time (depending on the input 
parameters). Three major job steps are required, with intermediate 
results stored on magnetic disk storage. The first step produces I (L) . 



159 

The second step produces I„(L). The third step computes T(L) and 
performs the integral transform /to produce P(e) . The final P(e) appears 
as printed and punched output as well as graphical display. 

The method employed to compute I (L) is the same as that discussed 
for the a search. The difference is that here we compute I (L) over a 
finer mesh of L values with consequently increased CPU time. For R = 
a normal CPU time for this step (for 95 L values) is approximately 50 
seconds. This time is doubled for cases of R ^ 0. We shall only repeat 
here that the accuracy in the computation of 1, (L) probably exceeds six 
significant figures. 

The evaluation of I„(L) involves a two-dimensional integration so 
that this step requires significantly more computer time than the first 
step. The case R = requires about 2.8 minutes of CPU time. This time 
is to be doubled if R / 0. The two-dimensional integration is carried 
out using a product of two one-dimensional rules (Trapezoidal and Simpson 

rules) . Also an algorithm developed especially for these half-quadrant 

48 
integrations is employed. Currently, the sum over k is terminated 

at k = 6. We obtain about three significant figures with k = 6 but it 

has been pointed out that for higher a. values (eg. a = /3) more terms may 

49 
be needed. The cost of evaluating these terms is considerable so 

that it is important to remember that I„(L) is expected to be a small 

correction to I (L) . That is to say, in the same spirit in which we 

linearized the Debye-HUckel pair correlation function, we now recognize 

that an error in I~(L) as large as ten per cent will not cause more than 

a two or three per cent error in the final P(e). (Normally I„(L) is less 

than twenty per cent of the magnitude of I (L).) This gives the 

justification for truncating the sum over k at k = 6. 






160 
The final step in this program reads as input the results of the 
previous step, computes T(L) in the second approximation, and evaluates 
the sine transform. This step requires one minute of CPU time. As 
we have indicated previously we must evaluate the following integral: 

CO 

P(e) - 2e7T~ 1 \ L T(L) sin(EL) dL f (H-3) 

o 

where 

T(L) = exp {-yL 2 + I 1 (L) + I^L)} . (H-4) 

The following approach was suggested by Coldwell. We approximate 
T(L) in the following manner: 

exp[f(x)] = E 9(x. , - - x) 6(x - x.) exp[a.x + b.] (H-5) 

l + l 11 i » 

where 

f - f 

i+l i f .x. , , - f . , .X. 

a l-*i + l-*l ' "i" 11 + 1 1 + 11 • (H-« 

X. , -, - X. 

1 + 1 1 
We have performed a piece-wise linear fit to the smoothe function in the 
exponent of T(L) . If we insert this expression into the integral of 
Equation (H-3) we obtain 



J 



I = \ x exp [f(x)] sin(ex) dx 
o 

x 



b. 1+1 

a .x 

x e sin(Ex) dx < (H-7) 






X. 

1 

This integral may be performed analytically with the result: 

. a.x 2 2 

b . l a . - e 

i r e "i 

I = 2 e {— — [(a.x - — -) sin ex 

i a . + e a . + e 

i i 



2a. 

(ex ■= -j — ) cos ex]} 

a . + e 



161 
X i+1 

(H-8) 



X l 



This computation is carried out using IBM's extended precision option 
to insure adequate numerical accuracy. 

As in all numerical transforms of this type, at large values of £ 
we encounter unphysical oscillations in the computed P(e). One inter- 
pretation of this phenomenon is that the numerical error is of the same 
order of magnitude as the exact value of the transform. With this in 
mind, we may regard the amplitude of these oscillations as an estimate 
of the error obtained in the numerical transform. Indeed, checks on 
the numerical accuracy of this technique indicate that, in the region 
where P(e) shows structure, we obtain about three significant figures of 
precision. Extension of the microfield table into the asymptotic region 
(large e) is carried out by the third computer program. 

The Asymptotic Microfield Program 

In the third program, the asymptotic microfield distribution function 
described in Section II is joined smoothly to the computed transform. 
Let us make the following definitions. The range of e values corresponding 
to the microfield peak we define to be Region I. The range of e values, 
e>100 (where we measure £ in units of e n , the Holtsmark field strength), 
we designate Region III. The intermediate region (Region II) contains 
the unphysical oscillations. We assume that for e>100, P(e) is given 
exactly by the asymptotic form. The approach taken here is to use the 

reliable results from Regions I and III to interpolate into Region II. 

5/2 
If we scale the data in Regions I and III by e (the inverse of 

the asymptotic Holtsmark distribution) and take the logarithm of the 






162 
result, we obtain a smooth curve. A fit to this curve is performed using 
a least-squares Monte Carlo Spline fitting routine. We simply invert 
the fit to obtain P(e) in Region II. In this manner we preserve the 
accuracy achieved in Region I and obtain a P(e) which makes a smooth 
transition to the asymptotic form. 

In order to make a statement about the numberical accuracy of this 
technique it is necessary to compare the fitted P(e) with the transform 
result in Region II. When these two results are plotted, several facts 
can be immediately noticed. First, as the two curves enter Region II, 
they overlie one another. Second, as e increases, the transform result 
begins to oscillate closely around the fitted P(e). These two facts tend 
to support our previous error estimate of a few per cent for P(e). In 
addition, the magnitude of P(e) in Region II is so small (^10~ ) that 
errors in this region are not likely to cause any difficulty in subsequent 
line profile calculations. 

The Stark Profile Program 

The fourth program generates the function J(u,e) discussed in 
Section I and carries out the microfield average over the static ions to 
produce the Stark broadened profile. In a final step, Doppler corrections 
are added by the convolution indicated in Equation (1-6) . The final 
profile appears as printed and punched output as well as graphical 
display. In addition to the appropriate microfield table, basic input 
to this program consists of the electron number density, the electron 
kinetic temperature, the nuclear charge of the radiator, and the principal 
quantum number n of the upper level of the desired Lyman transition. 
The CPU times required are 2.1 minutes for Lyman-a and 6.8 minutes for 






163 

Lyman- a . Most of this time is consumed by the matrix inversions required 

to produce J(w,e). The matrices which must be inverted are of order 

2 

n . 

u 

Preliminary computations of the electron broadening function r(Aw) 

for the cases studied in this work indicated that T (Aw) is smaller in 

Re 
3 
magnitude than Fj (Aw) by a factor of vl_0 . Therefore the program does 

not compute r (Aw) and it is assumed to be negligible. This causes 

most of the shift and asymmetry of the resulting line profile to vanish. 

We mention here that it would be straightforward task to include T (Aw). 

Re 

However, by neglecting r (Aw) we achieve a savings of approximately 3 

minutes of computer time. 

Another feature of the line profile calculation is an approximate 

treatment of electron correlations. In order to approximate the effect of 

electron correlations, we modify our ideal gas result for T T (Aw) such 

Im 

that, 

F Im (Aw i V = r im ( V ' < H ~ 9) 

This procedure is suggested by Figure 1 of Reference 22. If we impose 
the cutoff not at w but at some fraction or multiple thereof, we find 
that the line center will be lowered or raised. However the changes are 
small and quite insensitive to the cutoff frequency. 

In the Classical Path Impact Theory, the operator corresponding 
to r Re (Aw) is also neglected. Also, that theory treats electron 
correlations by a cutoff procedure yielding a result similar in form 
to that given in Equation (H-9.) . These two facts appear to give the 
present theory the form of a quantum mechanical analog to the Classical 
Path Impact Theory. 






164 
J(oi,e) is given by Equation (1-95): 



J(o),e) = -it Id I D, K , p., , (H-10) 

. . i 11 11 1 

n 
where D and p are real. Now 

<£. " A 1± . + IB.., , (H-ll) 

where from Equations (1-96) and (1-97) we have (after letting F (Aco) 
vanish) : 

A , -5 , {Aoj - 4 -g- n q.e} . (H-12) 

ii ii z nz u i ' 

-v -> 
B.., = K" 2 r im (Aa>) ^ l lin '* ±n±t ■ (H-13) 

We have 



K. , = C. , + iF. ., (H-14) 

ii n n' ' v ' 

such that [A + IB] [C + iF] = JL , and JL is the unit matrix. Now J(tD,e) 
reduces to 

J(oj,e) » — rr I D. , . F.., p., . (H-15) 

... ii ii i 
n 

From Reference 19 we have: 

F.., = -(B + AB _1 A)" 1 .., . (H-16) 

ii ii ' 

A final form for J(u),e) is given by 

J(a),e) = tt~ Z D. t . (B + AB A)"*, p ., , (H-17) 

ii' x 

This expression makes it obvious that for every value of Aid and e we must 

invert a matrix. The matrix inversion is carried out by an IBM-supplied 

subroutine MINV which has been modified to employ extended precision 



165 

arithmetic. 

The atomic factor in Eauation (H-5) is evaluated in Appendix G. . 

3 
The n distinct 3-j symbols required are computed by a subroutine given 

in Reference 33. T (Aid) is calculated in Appendix F. After J(u>,e) 

has been computed, the integration over the microfield distribution 

function P(e) is performed by a Trapezoidal Rule formula. 

In order to assess the numerical precision in the computation of 
J(o),e) we consider the last three equations in Section I. Starting with 
Equation (1-97) we note: (1) as discussed in Appendix F, T (Aw) is 
computed using a 40 point Gauss-Laguerre quadrature formula — numerical 
tests indicate that we obtain at least six figure accuracy; (2) in the 
computation of the atomic factor, we essentially are using single 
precision arithmetic (8 significant figures) to perform an integer 
arithmetic calculation. We obtain at least six figure accuracy in 
calculating H(Aa)) as given in Equation (1-97) . Now consider Equation 
(1-96). In the present work, all numerical operations in the production 
of R (w) , the effective radiator resolvent operator, (including the matrix 
inversion) are carried out using IBM's extended precision option (33 
significant figures). We retain six figure accuracy. In the case of 
Equation (1-95), argument (2) above holds for the operator D. The result 
is that we obtain at least six figure accuracy in the calculation of 
J(oi, e) . 

There are two possible checks on the accuracy of the microfield 

integration. The first check is to halve the integration interval and 

double the number of points. When this check was applied to a test case, 

we obtained ^4 place agreement between the two results. A second check 

involves extending the limit of integration. This check applied to a 

worst case (slowly decaying P(e), e.g., T = 0.25) yielded roughly 3 

R 






166 

place agreement. The result is that we obtain 3 significant figure 
accuracy for the Stark profiles. Actually the precision for the 
Doppler-corrected profiles is slightly better than this due to the 
"smoothing" effect of the Doppler convolution. 

Our conclusion is that the numerical error present in the final 
line profile is almost entirely due to the numerical error in the 
microfield functions. This allows us to set a rather conservative 
error bar on the final line profiles of a few per cent. 






APPENDIX I 



TABLES OF THE ELECTRIC MICROFIELD 
DISTRIBUTION FUNCTION P(e) 



In this appendix we present tables of the electric microfield 
distribution function P(e) computed using the numerical procedures 
discussed in Appendix H. The various parameters are defined as follows: 
A - The plasma parameter (= r /A ) . The value of this parameter 
is determined by the temperature and density of the electron 
perturbers. 

R - The ratio of the density of the charge z specie of ion perturber 
to the density of the charge z specie. 

TEMP RATIO - The ratio of the electron kinetic temperature to the 
ion kinetic temperature. 

CHARGE AT ORIGIN - The net charge of the hydrogenic radiator 
(= z-1). 

zl,z2 - The charges of the ion perturbers. 

2 
e is expressed in units of e (= e/r ) . 



167 






168 



ELECTRIC MICRQFIELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURB^RS 

A= 0.2000 R= 0,0 TEMP RATIQ= 1.00 

CHARGE AT ORIGlN= 9*00 Z 1= 1,00 Z2= 9.00 

E P(E) 

0. 10E 00 

0.20E 00 

0.30E 00 

0.40E 00 

0.50E 00 

0.60E 00 

0.70E 00 

0.80E 00 

0. 90E 00 

0. 10E 01 

0, I IE 01 

0.1 2E 01 

0. 13E 01 

0. 14E 01 

0.15E 01 

0. 16E 01 

0. 17E 01 

0, 18E 01 

0. 19c 01 

0.20E 01 

0.25E 01 

0.3 0E 01 

0.35E 01 

0.40E 01 

Q.45E 01 

0.50E 01 0.29671E-01 

0.60E 01 0.17142E-01 

0.7QE 01 0.10789E-01 

0.80E 01 0.72993E-02 

0.90E 01 0.52378E-02 

0.10E 02 0.39335E-02 

0.12E 02 0.24105E-02 

0.14E 02 0.15585E-02 

0.16E 02 0.10512E-02 

0.18E 02 0.73717E-03 

0.20E 02 0.53550E-03 

0.22E 02 0.40156E-03 

0.24E 02 0.30975E-03 

0.26E 02 0.24490E-03 

0.28E 02 0.19777E-03 

0-30E 02 0.16255E-03 

0.35E 02 0.10503E-03 

0.40E 02 0.71638E-04 

0.45E 02 0.50974E-04 

0.50E 02 0.37507E-04 

0.60E 02 0.21932E-04 

0.70E 02 0.13850E-04 

0.80E 02 0.92562E-05 

0.90E 02 0.64616E-05 

0. 10E 03 0.46688E-05 



0, 


82795E- 


-02 


= 


3 2355E- 


-01 


0, 


70015E- 


-0 1 


0, 


1 1 793E 


00 


0. 


1 7206E 


00 


0. 


2281 7E 


00 


0. 


2 8 22 7E 


00 


0. 


33099E 


00 


0< 


37184E 


00 


0< 


40329E 


00 


0* 


42471E 


00 


0, 


4 3629E 


00 


0. 


43883E 


00 


0. 


43352E 


00 


0, 


42178E 


00 


0. 


40507E 


00 


0« 


38481E 


00 


Q< 


3G224E 


00 


0< 


33844E 


00 


0« 


3 1427E 


00 


Or 


20551E 


00 


0. 


1 31 16E 


00 


= 


85068E- 


•01 


Q« 


57464E- 


-0 1 


0. 


40516E- 


-01 






169 



ELECTRIC MICRQFIELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURB5RS 



A= 0,4000 R= 0.0 

CHARGE AT ORIGIN= 9.00 



TEMP RATIO= 1.00 

Zl= 1.00 Z2= 9.00 



P(E) 



0.10E 00 

0.20E 00 

0.30E 00 

0.40E 00 

0.50E 00 

0.60E 00 

0.70E 00 

0.80E 00 

0.90E 00 

0. 10E 01 

0. HE 01 

0. 12E 01 

0. 13E 01 

0.14E 01 

0. 15E 01 

0.16E 01 

0. 17E 01 

0. 18E 01 

0, 19E 01 

0.20E 01 

0.25E 01 

0.30E 01 

0.3SE 01 

Q.40E 01 

0.45E 01 

0.50E 01 

0,60E 01 

0.70E 01 

0.80E 01 

0.90E 01 

0.10E 02 

0.12E 02 

0. 14E 02 

0. 16E 02 

0. 18£ 02 

0.20E 02 

0.22E 02 

0.24E 02 

0.2&E 02 

0.28E 02 

0.30E 02 

0.35E 02 

0.40E 02 

0.45E 02 

0.50E 02 

0.60E 02 

0.70E 02 

0.80E 02 

0.90E 02 

0. 10E 03 



0. 19933E- 


-01 


0.76325E- 


-01 


0. 1S982E 


00 


0.25750E 


00 


0.35582E 


00 


0. 


.44323E 


00 


0.51 181E 


00 


0.55773E 


00 


0, 


.58074E 


00 


0.S8321E 


00 


0.56899E 


00 


0.54242E 


00 


0.50765E 


00 


0.46826E 


00 


0.42706E 


00 


0< 


.38613E 


00 


0, 


.34688E 


00 


0, 


31 01 8E 


00 


0, 


.2 7649E 


00 


0. 


.24600E 


00 


= 


1 3715E 


00 


0. 


.79675E- 


-01 


0, 


48503E- 


01 


Q< 


3 1248E- 


-01 


0< 


2 1209E- 


-0 1 


0« 


1 4989E- 


-01 


0. 


80386E- 


■02 


0, 


45998E- 


-02 


0, 


2 794 3E- 


-02 


Oc 


17931E- 


-02 


0< 


12092E- 


02 


0, 


62569E- 


03 


= 


37178E- 


-03 


0.. 


24365E- 


03 


0^ 


I6914E- 


-03 


Oc 


12035E- 


03 


0, 


87 183E- 


-04 


= 


64236E- 


-04 


0« 


48092E- 


■04 


0. 


3&551E- 


04 


0„ 


28174E- 


•04 


0. 


15539E- 


■04 


Oc 


91622E- 


■05 


0. 


56849E- 


■05 


Go 


36742E- 


•05 


0. 


16876E- 


•05 


0. 


8 5331 E- 


■06 


0, 


46362E- 


06 


0, 


26642E- 


•06 


0,., 


160 13E- 


06 



170 



ELECTRIC MICROFIELO DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTUR3ERS 



A= 0.6000 R= 0.0 

CHARGE AT ORIGIN^ 9*00 



TEMP RATIO= I. 00 

Zl= 1.00 Z2 = 9.00 



PIE) 



Oc 


10E 


00 


= 


20E 


00 


0. 


30E 


00 


0. 


40E 


00 


0* 


5 0E 


00 


0. 


60E 


00 


Oo 


70E 


00 


= 


80E 


00 


Oo 


90E 


00 


Oo 


10E 


01 


0., 


1 IE 


01 


0, 


12E 


01 


0» 


13E 


Oi 


0* 


14E 


5 


Oo 


15E 


01 


Oo 


16E 


01 


Oo 


I 7E 


Oi 


Oo 


18E 


oa 


0. 


19E 


01 


0. 


20E 


01 


Oo 


25E 


01 


0. 


30E 


0! 


0. 


35E 


01 


Oo 


40E 


01 


0. 


45E 


01 


0. 


50E 


oa 


Oo 


60E 


01 


0. 


70E 


01 


Oo 


80E 


oi 


Oo 


90E 


01 


0, 


10E 


02 


Oo 


12E 


02 


0. 


14E 


02 


0. 


16E 


02 


0, 


18E 


02 


0. 


20E 


02 


0. 


22E 


02 


0. 


24E 


02 


0. 


26E 


02 


0. 


28E 


02 


0. 


30E 


02 


0. 


35E 


02 


0. 


40E 


02 


0. 


45E 


02 


0. 


50E 


02 


0. 


60E 


02 


0. 


70E 


02 


0. 


80E 


02 


0. 


90E 


02 


0. 


10E 


03 



0.5Q061E-01 
0.18414E 00 
0.36197E 00 
0.53756E 00 
0.67584E 00 
0.76003E 00 
0.78978E 00 
0.77485E 00 
0.72874E 00 
0.66441E 00 
0.59217E 00 
0.51927E 00 
0.45019E 00 
0.38734E 00 
0.33 169E 00 
0.28332E 00 
0.24180E 00 
0.20646E 00 
0.17652E 00 
0.15124E 00 
0.72915E-01 
0.38276E-01 
0.21727E-01 
0. 1 3077E-01 
0.8 1010E-02 
0.51674E-02 
0.22912E-02 
0.1 1280E-02 
0.6101 IE-03 
0.35875E-03 
0.22692E-03 
0.10834E-03 
0.6 1023E-04 
0.37269E-04 
0.23137E-04 
0. 14510E-04 
0.92342E-05 
0.59900E-05 
0.39741E-05 
0.26941E-05 
0.18622E-05 
0.79448E-06 
0.36830E-06 
0. 18222E-06 
0.95030E-07 
0.29264E-07 
0. 10237E-07 
0.39452E-08 
0.16415E-08 
0.72675E-09 



171 



ELECTRIC MICROFIELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERIUR3ERS 

A= 0.8000 R= 0.0 TEMP RAT10= 1.00 
CHARGE AT ORIGIN^ 9.00 Zl = 1.00 Z2= 9.00 

E P(E> 

0. 10E 00 0.12237E 00 

0.20E 00 0.41742E 00 

0.30E 00 0.73441E 00 

0.40E 00 0.95661E 00 

0.50E 00 0.10474E 01 

0.60E 00 0.10281E 01 

0.70E 00 0.93966E 00 

0.80E 00 0.81936E 00 

0.90E 00 0.69277E 00 

O.IOE Oi 0.57426E 00 

0.11E 01 0.47032E 00 

0.12E 01 0.38264E 00 

0. 13E 01 0.31040E 00 

0. 14E 01 0.25178E 00 

0. 15E 01 0.20455E 00 

0.16E 01 0.lfc»669E 00 

0. 17E 01 0.13636E 00 

0. 18E 01 0. 1 1204E 00 

0. 19E 01 0.92505E-01 

0.20E 01 0.76738E-01 

0.25E 01 0.32482E-01 

0.30E 01 0.15390E-01 

0.35E 01 0.77776E-02 

0.40E 01 0.41S04E-02 

0.45E 01 0.23258E-02 

0.50E 01 0.13616E-02 

0.60E 01 0.52827E-03 

0.70E 01 0.23S64E-03 

0.80E 01 0.11756E-Q3 

0.90E 01 0.64287E-04 

O.IOE 02 0.37953E-04 

0.12E 02 0.15666E-04 

0.14E 02 0.73255E-05 

0. 16E 02 0.35754E-05 

0.18E 02 0.18041E-05 

0.20E 02 0.93957E-06 

0.22E 02 0.50419E-06 

0.24E 02 0.27831E-06 

0.2GE 02 0.15776E-06 

0.28E 02 0.91682E-07 

0.30E 02 0.54532E-07 

0.35E 02 0.16307E-07 

0.40E 02 0.54218E-08 

0.45E 02 0.19609E-08 

0.5QE 02 0.75974E-09 

0.60E 02 0.13436E-09 

0.70E 02 0.28175E-10 

0.80E 02 0.67364E-11 

0.90E 02 0.17885E-11 

O.IOE 03 0.5 1737E-12 






172 



ELECTRIC MICROFIELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTUR3ZRS 



A= 0.2000 R= 0, 

CHARGE AT ORIGIN= 17.00 



TEMP RATIO= 1.00 

Zl= I. 00 22= 17.00 



P(E) 



0. 10E 00 

0.20E 00 

0.30E 00 

0.40E 00 

0.50E 00 

0.60E 00 

0.70E 00 

O. 8 0E 00 

0.90E 00 

0. 10E 01 

0. 1 IE 01 

0. 12E 01 

0.13E 01 

0.14E 01 

0. 15E 01 

0. 16E 01 

0.17E 01 

0. 13E 01 

0. 19E 01 

0.20E 01 

0.25E 01 

0.30E 01 

0.35E 01 

0.40E 01 

0.45E 01 

0.50E 01 

0.60E 01 

0.70E 01 

0.80E 01 

0.90E 01 

0. 10E 02 

0. 12E 02 

0. 14E 02 

0. 16E 02 

0.18E 02 

0.20E 02 

0.22E 02 

0.24E 02 

0.26E 02 

0.28E 02 

0.30E 02 

0.35E 02 

O.40E 02 

0.45E 02 

0.50E 02 

0.60E 02 

0. 70E 02 

0.80E 02 

0.90E 02 

0.10E 03 



0.9S228E-02 
0.37158E-01 
0.80215E-01 
0.13465E 00 
0.19562E 00 
0.25808E 00 
0.3 1733E 00 
0.36968E 00 
0.41225E 00 
0.44355E 00 
0.46313E 00 
0.47149E 00 
0.46980E 00 
0.45964E 00 
0.44278E 00 
0.42099E 00 
0.39590E 00 
0.36892E 00 
0.34I22E 00 
0.31371E 00 
0.19583E 00 
0.12025E 00 
0.75688E-01 
0.49866E-01 
0.34268E-01 
0.24542E-01 
0.13796E-01 
0.83781E-02 
0.53663E-02 
0.36029E-02 
0.25254E-02 
0.13890E-02 
0.36449E-03 
0.58938E-03 
0.42614E-03 
0.31716E-03 
0.24048E-03 
0.18514E-03 
0.1 4534E-03 
0.1 1575E-03 
0.93423E-04 
0.57539E-04 
0.37575E-04 
0.25666E-04 
0. 18169E-04 
0.98870E-05 
0.58438E-05 
0.36723E-05 
0.24195E-05 
0.16551E-05 



173 



ELECTRIC MICROFIELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTJR3ERS 

A= 0.4000 R= 0.0 TEMP RATIO= I. 00 

CHARGE AT ORIGIN^ 17.00 Zl = 1.00 12 = 17.00 

E P(E> 

0.10E 00 0.28835E-01 

0.20E 00 0.10951E 00 

0.30E 00 0.22627E 00 

0.40E 00 0.35799E 00 

0.50E 00 0.48363E 00 

0.60E 00 0.58673E 00 

0.70E 00 0.65769E 00 

0.80E 00 0.69383E 00 

0.90E 00 0.69796E 00 

0. 10E 01 0.67612E 00 

0.11E 01 0.63564E 00 

0. 12E 01 0.58360E 00 

O. 13E 01 0.52597E 00 

0. 14E 01 0.46730E 00 

0. 15E 01 0.4 1072E 00 

0.16E 01 0.35815E 00 

0. 17E 01 0.31061E 00 

0. 18E 01 0.26843E 00 

0. 19E 01 0.23153E 00 

0.20E 01 0.19958E 00 

0.25E 01 0.96666E-01 

0.30E 01 0.50139E-01 

0.35E 01 0.27997E-01 

0.40E 01 0.16809E-01 

0.45E 01 0.10658E-01 

0.50E 01 0.70776E-02 

0.60E 01 0.34344E-02 

0.70E 01 0.18018E-02 

0.80E 01 0.10092E-02 

0.90E 01 0.60027E-03 

0.10E 02 0.37715E-03 

0.12E 02 0.1717SE-03 

0.14E 02 0.91379E-04 

0.16E 02 0.54402E-04 

0.18E 02 0.34735E-04 

0.20E 02 Q.22793E-04 

0.22E 02 0.14962E-04 

0.24E 02 0.10010E-04 

0.26E 02 0.68513E-05 

0.28E 02 0.47876E-05 

0.30E 02 0.34090E-05 

0.35E 02 0.15630E-05 

0.40E 02 0.77485E-06 

0.4SE 02 0.40847E-06 

0.50E 02 0.22627E-06 

0.60E 02 0.78005E-07 

0.70E 02 0.30289E-07 

0.80E 02 0.12870E-07 

0.90E 02 0.58705E-08 

0. 10E 03 0.28358E-08 



174 






ELECTRIC MICROFIELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBcRS 

A= 0.6000 R= 0.0 TEMP RATIQ= 1.00 
CHARGE AT ORIGIN^ 17.00 Zl= I. 00 Z2= 17.00 

E P(E) 

0.10E 00 0.87047E-01 

0.20E 00 0.31204E 00 

0.30E 00 0.58882E 00 

0.40E 00 0.82879E 00 

0.50E 00 0.97782E 00 

0.60E 00 0.10245E 01 

0.70E 00 0.98700E 00 

0.80E 00 0.89527E 00 

0.90E 00 0.77760E 00 

0. 10E 01 0.65488E 00 

O.ilE 01 0.53982E 00 

0. 12E 01 0.43866E 00 

0.13E 01 0.35331E 00 

0. 14E 01 0.28320E 00 

0. 15E 01 0.22662E 00 

0. 16E 01 0. 18145E 00 

0« 17E 01 0. 14561E 00 

0. 18E 01 0.1 1725E 00 

0.19E 01 0.94821E-01 

0.20E 01 0.77057E-01 

0.25E 01 0.29511E-01 

0.30E 01 0.12954E-01 

0. 35E 01 0.65264E-Q2 

0.40E 01 0.37288E-02 

0.45E 01 0.23715E-02 

0.50E 01 0.15980E-02 

0.60E 01 0.74376E-03 

0.70E 01 0.35938E-03 

0.80E 01 0.18001E-03 

0.90E 01 0.93331E-04 

0.10E 02 0.50016E-04 

0.12E 02 0.15767E-04 

0.14E 02 0.5S728E-05 

0.1 6E 02 0.2 1825E-05 

0. 18E 02 0.93600E-06 

0.20E 02 0.43440E-06 

0.22E 02 0.21561E-06 

0.246 02 0.11311E-06 

0.26E 02 0.61975E-07 

0.28E 02 0.35056E-07 

0.30E 02 0.20323E-07 

0.35E 02 0.57179E-08 

0.40E 02 0.17985E-08 

0.45E 02 0-61783E-09 

0.50E 02 0.22812E-09 

0.60E 02 0.36929E-10 



0. 70E 02 0.71466E-1 I 

0.80E 02 ' 0.15869E-H 

0.90E 02 0.39320E-12 

0.10E 03 0.10660E-12 



0.80E 02 ' 0.15869E-11 

0.90E 02 0.39320E-12 






175 



ELECTRIC MICROFtELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBERS 

A= 0.8000 R- 0.0 TEMP RATION I. 00 
CHARGE AT ORIGIN= 17.00 Zl = 1.00 Z2 = 17.00 

E PIE) 

O. 10E 00 0.24017E 00 

0.20E 00 0.7692SE 00 

0.30E 00 0.12320E 01 

C.40E 00 0.14314E 01 

0.50E 00 0.13S16E 01 

0.60E 00 0.1 1891E 01 

0.70E 00 0.9S200E 00 

0.80E 00 0.72853E 00 

0.90E 00 0.54282E 00 

O.IOE 01 0.39835E 00 

O.llE 01 0.29057E 00 

0. 12E 01 0.21 173E 00 

0.13E 01 0.15468E 00 

0. 14E 01 0. 1 1 377E 00 

0. 15E 01 0.84086E-01 

0. 16E 01 0.62798E-01 

0.17E 01 0.47238E-01 

0.18E 01 Q.35784E-01 

O. 19E 01 0.27502E-01 

0.20E 01 0.21 122E-01 

0.25E 01 0.61281E-02 

0.30E 01 0.23034E-02 

0.35E 01 0.1 1 119E-02 

0.40E 01 0.63823E-03 

0.45E 01 0.37731E-03 

0.50E 01 0.22775E-03 

0.60E 01 0.84627E-04 

0.70E 01 0.32703E-04 

0.80E 01 0.13126E-04 

0.90E 01 0.54647E-05 

O.IOE 02 0.23568E-05 

0.12E 02 0.48479E-06 

0.14E 02 0.113Q6E-06 

0. 16E 02 0.29582E-07 

0.186 02 0.85947E-08 

0.20E 02 0.27439E-08 

0.22E 02 Q.95263E-09 

0.24E 02 0.35593E-09 

0.26E 02 0.14164E-09 

0.28E 02 0.59404E-10 

0.30E 02 0.25988E-10 

0.35E 02 0.37381E-11 

0.40E 02 Q.62702E-12 

0.45E 02 0. 1 191 1E-12 

0.50E 02 0.25071E-13 

0.60E 02 0.14225E-14 

0.70E 02 0.1Q465E-15 

0.80E 02 0.94251E-17 

O. 90E 02 0.99924E-18 

O.IOE 03 0.12121E-I8 



176 



ELECTRIC MICROFIELO DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURB=RS 

A= 0.2830 R= 0.0 TEMP RATIQ= 1.00 

CHARGE AT ORI GI N= 9,00 Zl~ 1.00 Z2= 9.00 

E PCE) 

0.10E 00 0.11729E-01 

0.20E 00 0.45525E-01 

0.30E 00 0.97467E-01 

0.40E 00 0.16178E 00 

0.50E 00 0.23179E 00 

0.60E 00 0.30092E 00 

0.70E 00 0.36351E 00 

0.80E 00 0.41540E 00 

0.9QE 00 0.45413E 00 

O.IOE 01 0.47887E 00 

0. I IE 01 0.49015E 00 

0.12E 01 0.48943E 00 

0.13E 01 0.47876E 00 

0. 14E 01 0.46039E 00 

0.15E 01 0.43653E 00 

0. 16E 01 0.40914E 00 

0.17E 01 0.37989E 00 

0. 18E 01 0.35010E 00 

0.19E 01 0.32075E 00 

0.20E 01 0.292S5E 00 

0.2SE 01 0.17886E 00 

0.30E 01 0.1096SE 00 

0.35E 01 0.69770E-01 

0,40E 01 0.46423E-01 

0.45E 01 0.32220E-01 

0.50E 01 0.23202E-01 

0.60E 01 0.13262E-01 

0.70E 01 0.81944E-02 

0.80E 01 0.53917E-02 

0.90E 01 0.37480E-02 

0.10E 02 0.27307E-02 

0.12E 02 0.16169E-02 

0. 14E 02 0.10505E-02 

0. 16E 02 0.71261E-03 

0.18E 02 0.50000E-03 

0.20E 02 0.36201E-03 

0.22E 02 0.26983E-03 

0.24E 02 0.20656E-03 

0.26E 02 0.16202E-03 

0.28E 02 0.12991E-03 

0.30E 02 0.10624E-03 

0.35E 02 0.68949E-04 

0.40E 02 0.48069E-04 

0.45E 02 0.34714E-04 

0.50E 02 0.25393E-04 

0-60E 02 0.14034E-04 

0.70E 02 0.80836E-05 

0.80E 02 0.48420E-05 

0.90E 02 0.30094E-05 

O.iOE 03 0.19364E-05 






177 



ELECTRIC MICROFIELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTUR3ERS 

A= 0.2630 R= 0.0 TEMP RATIQ= 1.00 
CHARGE AT ORIGIN^ 17.00 Zl = 1.00 Z2 = 17.00 

E P(E) 

0.10E 00 0.14925E-01 

0.20E 00 0.57726E-01 

0.30E 00 0.12287E 00 

0.4QE 00 0.20233E 00 

0.50E 00 0.28700E 00 

0.60E 00 0.36819E 00 

0.70E 00 0.43876E 00 

0.80E 00 , 0.49388E 00 

0.90E 00 0.53115E 00 

O.IOE 01 0.55038E 00 

0. HE 01 0.55309E 00 

0.I2E 01 0.S4186E 00 

0. 13E 01 0.5 1981E 00 

0.14E 01 0.49Q06E 00 

0.15E 01 0.45549E 00 

0. 16E 01 0.4 1850E 00 

0. 17E 01 0.38099E 00 

0.18E 01 0.34436E 00 

0. 19E 01 0.30955E 00 

0.20E 01 0.27714E 00 

0.25E 01 0.15579E 00 

0.30E 01 0.89203E-01 

0.35E 01 0.53722E-01 

0.40E 01 Q.34172E-01 

0.4SE 01 0.22833E-01 

O.SOE 01 0.15907E-01 

0.60E 01 0.85994E-02 

0.70E 01 0.50563E-02 

0.80E 01 0.31854E-02 

0.90E 01 0.21320E-02 

O.IOE 02 0.15032E-02 

0. 12E 02 0.84441E-03 

0.14E 02 0.5276&E-03 

0. 16E 02 0.34548E-03 

0. 18E 02 0.23315E-G3 

0.20E 02 0.16189E-03 

0.22E 02 0. I 1S45E-03 

0.24E 02 0.84402E-04 

0.26E 02 0.63149E-04 

0.28E 02 0.48266E-04 

0.30E 02 0.37fjl8E-04 

0.35E 02 0.21720E-04 

0.40E 02 0.13628E-04 

0.45E 02 0.90368E-05 

0.50E 02 0.61619E-05 

0.60E 02 0.29770E-05 

0.70E 02 0.15079E-05 

0.80E 02 0.79879E-06 

0.90E 02 0.44141E-06 

O.IOE 03 0.25383E-06 



178 



ELECTRIC MICROFIELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTUR3ERS 

A= 0.1362 R= 0.0 TEMP RATIQ= I. 00 

CHARGE AT ORIGIN= 17.00 Z 1= I. 00 Z 2- 17.00 

E P(EJ 

0.69354E-02 
0.27188E-01 
0.59163E-01 
0.10040E 00 
0.14787E 00 
0.19 826E 00 
0.24831E 00 
0.29512E 00 
0.33632E 00 
0.37022E 00 
0.39582E 00 
0.41281E 00 
0.42143E 00 
0.42236E 00 
0.41660E 00 
0.40530E 00 
0.38966E 00 
0.37084E 00 
0.34990E 00 
0.32776E 00 
0.22042E 00 
0.14173E 00 
0.92188E-01 
0.62Q17E-01 
0.43359E-01 
0.31434E-01 
O.I8138E-01 
0. 1 1373E-01 
0.77231E-02 
0.56134E-02 
0.43160E-02 
0.20816E-02 
0.20932E-02 
0. 1S587E-02 
0. 1 1827E-02 
0.91365E-03 
0.71772E-03 
0.57276E-03 
0.46384E-03 
0.38080E-03 
0.31658E-03 
0.20936E-03 
0. 14639E-03 
0. 10647E-03 
0.79232E-04 
0.45488E-04 
0.27279E-04 
0.1 7045E-04 
0. 1 1071E-Q4 
0.7455BE-05 



0, 


10E 


00 


= 


20E 


00 


Oo 


3 0E 


00 


Oc 


4 0E 


00 


= 


50E 


00 


Or 


60E 


00 


Q« 


70E 


00 


0. 


8 0E 


00 


0, 


90E 


00 


0„ 


10E 


01 


0. 


1 IE 


01 


Q* 


12E 


01 


Oo 


13E 


01 


0* 


14E 


01 


0, 


15E 


01 


Oo 


16E 


01 


0. 


17E 


01 


Oo 


18E 


01 


Oo 


19E 


01 


Oo 


20E 


01 


0. 


25E 


01 


0. 


30E 


01 


Oo 


35E 


01 


Oo 


40E 


01 


0. 


45E 


01 


0, 


50E 


01 


Oo 


60E 


01 


0. 


70E 


01 


Oo 


80E 


01 


Oo 


90E 


01 


0. 


10E 


02 


Oo 


12E 


02 


0. 


14E 


02 


0, 


16E 


02 


0. 


18E 


02 


0. 


20E 


02 


0. 


22E 


02 


0. 


24E 


02 


0. 


26E 


02 


0. 


2 8E 


02 


0. 


30E 


02 


0. 


35E 


02 


0. 


40E 


02 


0. 


45E 


02 


0. 


50E 


02 


0. 


60E 


02 


0. 


70E 


02 


0. 


80E 


02 


0. 


90E 


02 


0. 


10E 


03 



179 



ELECTRIC MICROFIELO DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTUR35RS 

A= 0.1731 R= 0.0 TEMP RATIO= 1.00 

CHARGE AT ORIGIN^ 9.00 Zl= I. 00 Z2= 9.00 

E P(E) 

0.75651E-02 
0.29603E-01 
0.64224E-01 
0.10855E 00 
0.15905E 00 
0.21198E 00 
0.26372E 00 
0.3H15E 00 
0.35184E 00 
0.38420E 00 
Q.40744E 00 
0.42150E 00 
0.42690E 00 
0.42462E 00 
0.4 1584E 00 
0.40189E 00 
0.38407E 00 
0.363S6E 00 
0.34144E 00 
0.31856E 00 
0.21207E 00 
0.1366IE 00 
0.89521E-01 
0.60752E-01 
0.42821E-01 
0.31258E-Q1 
0. 1 8068E-01 
0.1 1586E-01 
0.81328E-02 
0.61632E-02 
0.49736E-02 
0.37012E-02 
0.30 146E-02 
0.25029E-02 
0.20842E-02 
0.1 7406E-02 
0.1 4578E-02 
0. 12244E-02 
0. 10313E-02 
0.87107E-03 
0.73776E-03 
0.49288E-03 
0.33480E-03 
0.231 14E-03 
0. 16211E-03 
0.83462E-04 
0.45535E-04 
0.26235E-04 
O. 15908E-04 
O. 1 01 16E-04 



Or 


10E 


00 


= 


2 0E 


00 


0. 


30E 


00 


= 


40E 


00 


Q« 


bOE 


00 


0. 


60E 


00 


Oc 


70E 


00 


3, 


80E 


00 


0. 


90E 


00 


Oc 


10E 


Oi 


Oc 


HE 


01 


0. 


12E 


01 


0. 


13E 


01 


0* 


14E 


01 


0. 


15E 


01 


0, 


16E 


01 


0* 


17E 


01 


Oo 


18E 


01 


o 


19E 


01 


Oo 


20E 


01 


0. 


25E 


01 


0. 


30E 


01 


0. 


35E 


01 


0. 


40E 


01 


0. 


45E 


01 


0. 


50E 


01 


0. 


60E 


01 


0. 


70E 


01 


0. 


80E 


Oi 


0. 


90E 


01 


0. 


10E 


02 


0. 


12E 


02 


0. 


14E 


02 


Q. 


16E 


02 


0. 


18E 


02 


. 


20E 


02 


0. 


22E 


02 


0. 


24E 


02 


0. 


26E 


02 


0. 


28E 


02 


0. 


30E 


02 


0. 


35E 


02 


0. 


40E 


02 


0. 


45E 


02 


0. 


50E 


02 


0. 


60E 


02 


0. 


70E 


02 


0. 


80E 


02 


0. 


90E 


02 


0. 


10E 


03 



180 



ELECTRIC MICROFIELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTUR3ERS 



A= 0,2246 R= 0.0 

CHARGE AT ORIGIN= 17,00 



Zl = 



TEMP RATIO= 1,00 

1.00 Z2 = 17.00 



PIE) 



0.1 OE 00 
0.20E 00 
0.30E 00 
0.40E 00 
O.SQE 00 
0.60E 00 
0.70E 00 
0.80E 00 
0.90E 00 
0.10E 01 
0. 1 IE 01 
0.12E 01 
0. 13E 01 
0. 14E 01 
0.15E 01 
0. 16E 01 
0. 17E 01 
0.1 8E 01 
0. 19E 01 
0.20E 01 
0.25E 01 
0.30E 01 
0.35E 01 
0.40E 01 
0.45E 01 
0.50E 01 
0.60E 01 
0.70E 01 
0.80E 01 
0.90E 01 
0.10E 02 
0.12E 02 
0. 14E 02 
0. 16E 02 
0.1 8E 02 
0.20E 02 
0.22E 02 
0.24E 02 
0.26E 02 
0.28E 02 
0.30E 02 
0.35E 02 
0.40E 02 
0.45E 02 
0.50E 02 
0.60E 02 
0.70E 02 
0.80E 02 
0.90E 02 
0. 10E 03 



0. 10832E-01 
0.42164E-01 
0.9Q6 96E-0 1 



0.15150E 
0.21874E 
0.28648E 
0.34940E 
0.40331E 
0.44S43E 
0.47443E 
0.49027E 
0.49392E 
0.48703E 
0.47162E 
0.44980E 
0.42356E 
0.39467E 
0.36459E 
0.33446E 
0.30515E 
O. 1 8474E 
0.1 1 103E 
0.69143E- 



00 
00 
00 
00 
00 
00 
00 
00 
00 
00 
00 
00 
00 
00 
00 
00 
00 
00 
00 
01 



0.451 17E-01 
0.30795E-01 
0.21 865E-01 
0.98212E-02 
0.51315E-02 
0.28833E-02 
0.17337E-02 
0.1 1 101E-02 
0.53878E-03 
0.31692E-03 
0.2 1719E-03 
0. 16670E-03 
0. 13776E-03 
0. 1 t 784E-03 
0.10 137E-0 3 
0.87375E-04 
0.75455E-04 
0.65286E-04 
0.45838E-04 
0.32557E-04 
0.23386E-04 
0. 16985E-04 
0.92500E-05 
0.52474E-05 
O.3O941E-05 
0. 18923E-05 
0.1 1977E-05 



181 



ELECTRIC MICROFIELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBzRS 

A= 0-121* R= 0,0 TEMP RATIO= 1.00 

CHARGE AT ORIGIN= 9.00 Z I- I. 00 Z2= 9.00 

E P(E) 



0. 


10E 


00 


0. 


2 0E 


00 


0. 


30E 


00 


0. 


4QE 


00 


0. 


50E 


00 


0. 


60E 


00 


0. 


70E 


00 


0» 


80E 


00 


0. 


90E 


00 


0. 


10E 


01 


0, 


1 IE 


01 


0. 


12E 


01 


0. 


13E 


OS 


0. 


14E 


0! 


0. 


15E 


01 


Oo 


16E 


01 


= 


17E 


01 


Oo 


18E 


01 


0. 


19E 


Oi 


0. 


20E 


01 


Oo 


25E 


01 


0. 


30E 


01 


0. 


35E 


01 


Q« 


40 E 


01 


0. 


45E 


01 


0. 


50E 


Oi 


= 


60E 


01 


0. 


70E 


01 


Oo 


80E 


01 


0. 


90E 


01 


Go 


10E 


02 


Go 


12E 


02 


. 


14E 


02 


Oo 


16E 


02 


Oo 


18E 


02 


0. 


20E 


02 


Oo 


22E 


02 


0. 


24E 


02 


0. 


26E 


02 


G„ 


28E 


02 


= 


30E 


02 


Oo 


35E 


02 


• 


4Q£ 


02 


0, 


45E 


02 


0, 


50E 


02 


O* 


60E 


02 


0. 


70E 


02 


Oo 


80E 


02 


0, 


90E 


02 


Oo 


10E 


03 



0. 


61 114E- 


-02 


0, 


23991E- 


01 


c< 


52327E- 


-01 


0. 


89088E- 


-OX 


0. 


131 74E 


00 


= 


17748E 


00 


0< 


22353E 


00 


0, 


26732E 


00 


0, 


.30672E 


00 


0« 


.34009E 


00 


0< 


.36642E 


00 


0, 


.38522E 


00 


0. 


, 39653E 


00 


0.40077E 


00 


0.39868E 


00 


0. 


,391 18E 


00 


0.37928E 


00 


0.36398E 


00 


0.34624E 


00 


0.32691E 


00 


0, 


■ 22780E 


00 


0.1S065E 


00 


0. 100 19E 


00 


0.68697E- 


-01 



0.48944E-01 
0.36270E-01 
0. 19896E-01 
0.1 1997E-01 
0.77649E-02 
0.53547E-02 
0.39054E-02 
0.23861E-02 
0. 16693E-02 
0. 12662E-02 
0. 10150E-02 
0.848I8E-03 
0.72B59E-03 
0.63453E-03 
0.55454E-03 
0.48E>63E-03 
0.42616E-03 
0.31016E-03 
0.228S6E-03 
0. I 7047E-03 
0.12865E-03 
0.75814E-04 
0.46660E-04 
0.29917E-04 
O. 19934E-04 
0. 13769E-04 



182 



ELECTRIC MICRQFIELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBERS 

A= 0.2000 R= 0.0 TEMP RATIQ= 0.25 

CHARGE AT ORIGIN^ 9.00 Zl= 1.00 Z2= 9.00 

E P(E I 



0. 


10E 


00 


0. 


20E 


00 


0. 


30£ 


00 


0. 


40E 


00 


0. 


5 0E 


00 


0. 


60E 


00 


0. 


70E 


00 


0. 


80E 


00 


0. 


90E 


00 


0. 


10E 


01 


0. 


i ie 


01 


0. 


I2E 


01 


0. 


13E 


01 


0* 


14E 


01 


0. 


15E 


01 


Oo 


16E 


01 


0. 


17E 


01 


0. 


18E 


01 


0. 


19E 


01 


0. 


20E 


01 


Oo 


25E 


01 


0. 


30E 


01 


Oo 


35E 


01 


0. 


40E 


01 


Oo 


45E 


01 


0. 


50E 


01 


0* 


60E 


01 


Oo 


70E 


01 


Oo 


80E 


01 


Oo 


90E 


01 


Oo 


10E 


02 


0, 


12E 


02 


Oo 


I4E 


02 


Oo 


16E 


02 


0, 


18E 


02 


Oo 


20E 


02 


0« 


22E 


02 


Oo 


24E 


02 


Oo 


26E 


02 


Oo 


28E 


02 


Oo 


3 0E 


02 


Oo 


35E 


02 


Oo 


40E 


02 


0, 


45E 


02 


Oc 


50E 


02 


0. 


60E 


02 


0. 


70E 


02 


0. 


80E 


02 


Oo 


90E 


02 


Oo 


10E 


03 



0, 


69169E- 


■02 


Oc 


2708SE- 


-01 


0. 


5 8 83 IE- 


-0 1 


0, 


99596E- 


01 


Oc 


14624E 


00 


0, 


19540E 


00 


G< 


24381E 


00 


0. 


2 8 863 E 


00 


0, 


32761.E 


00 


0, 


35922E 


00 


0« 


38266E 


00 


0< 


39776E 


00 


0. 


40492E 


00 


0< 


.40490E 


00 


0, 


39874E 


00 


0< 


38758E 


00 


0< 


37256E 


00 


0, 


.35478E 


00 


Q c 


.33520E 


00 


Q< 


.31464E 


00 


0« 


-21571E 


00 


0< 


, 14264E 


00 


0. 


• 95532E- 


-01 



0.66001E-01 
0.47220E-01 
0.34918E-01 
0.208S7E-01 
0.13455E-01 
0.93047E-02 
0.67955E-02 
0.51675E-02 
0.32821E-02 
0.23002E-02 
0. 17232E-02 
0. 13429E-02 
0. 10731E-02 
0.86865E-03 
0.70815E-03 
0.58118E-03 
0.48Q12E-03 
0.39918E-03 
0.25849E-03 
0. 17367E-03 
0.12077E-03 
0.86731E-04 
0.48812E-04 
0.30391E-04 
0.20550E-04 
0. 14816E-04 
0.11 180E-04 



183 



ELECTRIC MICRQFIELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTUR8ERS 

A= 0.2000 R= 0.0 TEMP RATI0= 0.50 

CHARGE AT ORIGIN= 9.00 Zl = I. 00 Z2 = 9.00 

E PIE) 



0. 


10E 


00 


0. 


2 0E 


00 


0. 


3 0E 


00 


0. 


40E 


00 


0o 


5 0E 


00 


0. 


60E 


00 


Oo 


7 0E 


00 


0. 


80E 


00 


Oo 


90E 


00 


Oo 


10E 


01 


= 


1 IE 


01 


0. 


12E 


01 


0. 


13E 


01 


Oo 


14E 


01 


0. 


15E 


01 


Oo 


16E 


01 


Oc 


17E 


01 


Oo 


I8E 


01 


0. 


19E 


oa 


0. 


20E 


01 


Oc 


25E 


01 


. 


30E 


01 


Oo 


35E 


01 


Oo 


4 0E 


01 


Oo 


45E 


01 


Oo 


SOE 


01 


0. 


60E 


Oi 


Oc 


70E 


01 


Go 


SOE 


01 


0. 


90E 


01 


Oo 


10E 


02 


Oo 


12E 


02 


0. 


14E 


02 


Oo 


16E 


02 


0. 


13E 


02 


0. 


20E 


02 


Oo 


22E 


02 


Oo 


24E 


02 


Oo 


26E 


02 


0. 


28E 


02 


Oo 


30 E 


02 


Oo 


35E 


02 


0. 


40E 


02 


Oo 


45E 


02 


0., 


50E 


02 


Oo 


60E 


02 


Oo 


70E 


02 


Oo 


80E 


02 


Oo 


90E 


02 


Oo 


10E 


03 



Oc 


73606E- 


-02 


0, 


28802E- 


-01 


Oo 


62484E- 


-01 


0« 


10561E 


00 


Oo 


15474E 


00 


0. 


2Q62SE 


00 


0, 


25662E 


00 


Oc 


30284E 


00 


0, 


34256E 


00 


0, 


37425E 


00 


0. 


39714E 


00 


= 


41 1 18E 


00 


0< 


41689E 


00 


= 


41516E 


00 


0, 


40717E 


00 


Oc 


39416E 


00 


0, 


37737E 


00 


0, 


35795E 


00 


0, 


33690E 


00 


0< 


31507E 


00 


0< 


21252E 


00 


0. 


138 78E 


00 


0, 


92072E- 


-01 



0.63146E-01 
0.44905E-01 
0.33029E-01 
0.19445E-01 
0. 12496E-01 
0.85725E-02 
0.61816E-02 
0.46384E-02 
0.28790E-02 
0. 19393E-02 
0.13850E-02 
0. 10247E-02 
0.77259E-03 
0.59203E-03 
0.46077E-03 
0.36397E-03 
0.29161E-03 
0.23681E-03 
0. I 4863E-03 
0. 10001E-03 
0.71377E-04 
0.53471E-04 
0.33265E-04 
0.22244E-04 
0. 15501E-04 
0. I 1 199E-04 
0.83479E-05 



184 



ELECTRIC MICROFIELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTUR3ERS 

A- 0.2000 R= 0.0 TEMP RATIO= I. 00 

CHARGE AT ORIGIN- 9.00 Zl= 1.00 22= 9.00 

E P(E) 

0.82510E-02 
0..32244E-01 
0.69784E-01 
0.11757E 00 
0.17159E 00 
0.22761E 00 
0.28167E 00 
0.33039E 00 
0.37129E 00 
0.40281E 00 
0.42433E 00 
0.43601E 00 
0.43065E 00 
0.43344E 00 
0.42178E 00 
0.40515E 00 
0.38493E 00 
0.36241E 00 
0.33863E 00 
0.31447E 00 
0.20S68E 00 
0.13U7E 00 
0.85474E-0I 
0.57804E-01 
0.40638E-01 
0.29597E-01 
0. 17131E-01 
0. 10842E-01 
0.73268E-02 
0.S2021E-02 
0.38397E-02 
0.22831E-02 
0. 14898E-02 
0.10314E-02 
0.73817E-03 
0.54258E-03 
0.40877E-03 
0.31501E-03 
0.24782E-03 
0. 19862E-03 
0.16186E-03 
O.i 0297E-03 
0.69598E-04 
0.49155E-04 
0.36063E-04 
0.21281E-04 
0.13679E-04 
0.92259E-05 
0.64576E-05 
0.46688E-05 



0, 


10E 


00 


0. 


2 0E 


00 


0. 


30E 


00 


0. 


40E 


00 


0. 


50E 


00 


0. 


60E 


00 


0. 


70E 


00 


0. 


80E 


00 


0. 


90E 


00 


0. 


10E 


01 


0. 


1 IE 


01 


0. 


12E 


01 


0. 


13E 


01 


0, 


14E 


01 


0. 


15E 


01 


0. 


16E 


01 


0. 


17E 


01 


0. 


18E 


01 


0. 


19E 


01 


0. 


20E 


01 


0. 


25E 


01 


0. 


30E 


01 


0. 


35E 


01 


0. 


40E 


01 


0. 


45E 


01 


0. 


50E 


01 


0. 


60E 


01 


0. 


70E 


01 


0. 


80E 


01 


Oo 


90E 


01 


0. 


10E 


02 


0. 


12E 


02 


0. 


14E 


02 


0* 


16E 


02 


0. 


18E 


02 


0. 


20E 


02 


Oo 


22E 


02 


Oo 


24E 


02 


Oo 


26E 


02 


Oo 


28E 


02 


0. 


30E 


02 


Oo 


35E 


02 


Oo 


40E 


02 


Oo 


45E 


02 


0, 


50E 


02 


= 


60E 


02 


Oo 


70E 


02 


Oo 


80E 


02 


= 


90E 


02 


Oo 


tOE 


03 



185 



IN 



ELECTRIC MICROFIELD O I S TR IBUT I ON FUNCTION 
A PLASMA CONTAINING MULTIPLY CHARGEO ION PERTUR8ERS 



A= 0.2000 R= 0.0 

CHARGE AT ORIGIN= 9.00 



Zl = 



TEMP 



1.00 



RATIO= 
Z2 = 



2< 

9< 



00 
00 



P«E) 



0. IOE 00 
0.2 0E 00 
0.30E 00 
0.40E 00 
0.50E 00 
0.6 0E 00 
0.70E 00 
0.80E 00 
0.90E 00 
0. IOE 01 
0. I IE 01 
0. 12E 01 
0.1 3E 01 
0. 14E 01 
0. 15E 01 
0.16E 01 
0.17E 01 
0. 18E 01 
0. 19E 01 
0.20E 01 
0.25E 01 
0.30E 01 
0.35E 01 
0.40E 01 
0.45E 01 
0.50E 01 
0.60E 01 
0.70E 01 
0.80E 01 
Q.90E 01 
0. IOE 02 
0.1 2E 02 
0.14E 02 
0. 16E 02 
0.1 8E 02 
0.20E 02 
0.22E 02 
0.24E 02 
0.26E 02 
0.28E 02 
0.30E 02 
0.35E 02 
0.40E 02 
0.45E 02 
0.50E 02 
0.60E 02 
0.70E 02 
0.80E 02 
0.90E 02 
0. IOE 03 



0.10082E-01 
0.39290E-01 
0.84664E-01 
0.14177E 00 
0.20532E 00 
0.26988E 00 
0.33051E 00 
0.38323E 00 
0.42531E 00 
0.45533E 00 
0.47303E 00 
0.47915E 00 
0.47507E 00 
0.46258E 00 
0.44353E 00 
0.41995E 00 
0.39336E 00 
0.36523E 00 
0.33670E 00 
0.30864E 00 
0.19073E 00 
0. I 1647E 00 
0.73431E-01 
0.48396E-01 
0.33312E-01 
0.23826E-01 
0.13373E-01 
0.82431E-02 
0.54385E-02 
0.37759E-02 
0.27278E-02 
0. 15555E-02 
0.96490E-03 
0.63921E-03 
0.44385E-03 
0.31878E-03 
0.23590E-03 
0.17936E-03 
0.13972E-03 
0.11 119E-03 
0.90 137E-04 
0.56766E-04 
0.37731E-04 
0.25798E-04 
0. 18090E-04 
0.95363E-05 
0.54575E-05 
0.33480E-05 
0.21741E-05 
0. 14756E-05 



186 



ELECTRIC MICROFIELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBERS 

A= 0-2000 R= 0.0 TEMP RATIO= 4.00 

CHARGE AT ORIGIN^ 9.00 Zl= I. 00 Z2 = 9.00 

E PtE) 

0. 10E 00 0.13971E-01 

0.20E 00 0.54165E-01 

0.30E 00 0.11574E 00 

0.40E 00 0.19157E 00 

0.50E 00 0.27344E 00 

0.60E 00 0.35330E 00 

0.70E 00 0.42428E 00 

0.80E 00 0.48142E 00 

0.90E 00 0.52195E 00 

0.10E 01 0.b4512E 00 

0.11E 01 0.55191E 00 

0.12E 01 0.54444E 00 

0. 13E 01 0.52550E 00 

0.14E 01 0.49808E 00 

0.I5E 01 0.46500E 00 

0. 16E 01 0.42875E 00 

0.17E 01 0.39136E 00 

0.18E 01 0.35437E 00 

0.19E 01 0.31887E 00 

0.20E 01 0.28S58E 00 

0.25E 01 0.15956E 00 

0.30E 01 0.9Q274E-01 

0.35E 01 0.53715E-01 

0.40E 01 0.33818E-01 

0.45E 01 0.22409E-01 

0.50E 01 0.15S07E-01 

0.60E 01 0.82225E-02 

0.70E 01 0.48242E-02 

0.80E 01 0.30438E-02 

0.90E 01 0.2Q269E-02 

0.10E 02 0.14088E-02 

0.12E 02 0.74952E-03 

0. 14E 02 0.43435E-03 

0. 16E 02 0.26946E-03 

0.18E 02 0.17633E-03 

0.20E 02 0.12004E-03 

Q.22E 02 0.84494E-04 

0.24E 02 0.61290E-04 

0.26E 02 0.45669E-04 

0.28E 02 0.3484SE-04 

0.30E 02 0.27136E-04 

0.35E 02 0.15551E-0* 

0.40E 02 0.94481E-05 

0.45E 02 0.59196E-05 

0.50E 02 0.3ai36E-05 

0.60E 02 0.17096E-05 

0.70E 02 0.84023E-06 

0.80E 02 0.44690E-06 

0.90E 02 0.25392E-06 

0.10E 03 0.1S214E-06 



187 



ELECTRIC MICROFIELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBERS 

A= 0.2000 R= 0.10E 00 TEMP RATIO= I. 00 

CHARGE AT ORIGIN^ 9.00 Zl= 1.00 Z2 = 9.00 

P(E) 

0.49883E-02 
0.19604E-01 
0.42837E-01 
0.73121E-01 
Q.10849E 00 
0.14678E 00 
0.I8580E 00 
0.22351E 00 
0.25820E 00 
0.28853E 00 
0.31359E 00 
G.33290E 00 
0.34636E 00 
0.35420E 00 
0.35687E 00 
0.35499E 00 
0.34924E 00 
0.34036E 00 
0.32903E 00 
0.31590E 00 
0.24019E 00 
0.17200E 00 
0.12137E 00 
0.86255E-01 
Q.62353E-01 
0.46013E-01 
0.26755E-01 
0.16649E-01 
0. 10981E-01 
0.76200E-02 
0.55222E-02 
0.32282E-02 
0.21 122E-02 
0. 14922E-02 
0. 1Q982E-02 
0.82073E-03 
0.62085E-03 
0.47521E-03 
0.36790E-03 
0.28799E-03 
0.22786E-03 
O. 13 263E-03 
0.81888E-04 
0.53325E-04 
0.36419E-04 
0. 19163E-04 
0. 11404E-04 
0.73385E-05 
0.4941 OE-05 
0.34523E-05 



0. 


10E 


00 


0. 


2 0E 


00 


0. 


30E 


00 


0. 


40E 


00 


0. 


50E 


00 


0. 


6 0E 


00 


0. 


7 0E 


00 


0. 


80E 


00 


0* 


9 0E 


00 


0. 


10E 


01 


0. 


HE 


Oil 


0. 


12E 


01 


0. 


I3E 


01 


0, 


14E 


01 


0. 


15E 


01 


0. 


16E 


01 


0. 


17E 


01 


0. 


18E 


01 


0. 


19E 


01 


0, 


20E 


01 


0„ 


25E 


01 


Oo 


30E 


01 


Oo 


3SE 


03 


0„ 


40E 


01 


Oo 


4SE 


01 


0. 


50E 


01 


0. 


60E 


01 


0. 


70E 


01 


0. 


80E 


01 


0. 


90E 


01 


0. 


10E 


02 


0. 


12E 


02 


Oo 


14E 


02 


0. 


16E 


02 


0« 


18E 


02 


0« 


20E 


02 


0. 


22E 


02 


0. 


24E 


02 


Oo 


26E 


02 


0.=- 


28E 


02 


0, 


30E 


02 


Oc 


35E 


02 


Oo 


40E 


02 


0, 


45E 


02 


0« 


50E 


02 


Oc 


60E 


02 


Oc 


70E 


02 


0. 


80E 


02 


0. 


90E 


02 


0, 


10E 


03 



188 



ELECTRIC MICROFIELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBERS 

A= 0.2000 R= 0.10E 13 TEMP RATIO= I. 00 

CHARGE AT ORIGIN= 9.00 Zl = I. 00 Z2= 9.00 

E P<E) 

0.34144E-02 
0.13463E-0 1 
0.29580E-01 
0.50874E-01 
0.76208E-01 
0.10429E 00 
0.13376E 00 
0.16330E 00 
0.19172E 00 
0.218O0E 00 
0.24135E 00 
0.26122E 00 
0.27730E 00 
0.28948E 00 
0.29785E 00 
0.30264E 00 
Q.30417E 00 
0.30282E 00 
0.29903E 00 
0.29321E 00 
0.24633E 00 
0.19237E 00 
Q.14569E 00 
0.10934E 00 
0.8225&E-01 
0.6241 1E-01 
0.37298E-01 
0.23475E-01 
0. 15486E-01 
0. 10664E-01 
0.76367E-02 
0.43329E-02 
0.27397E-02 
O. 187QIE-02 
0. 13371E-02 
0.98094E-03 
0.72976E-03 
0.54966E-03 
0.41901E-03 
0.32314E-03 
0.25200E-03 
0.I4174E-03 
0.84725E-04 
0.53478E-04 
0.35417E-04 
0. 17488E-04 
0.96901E-05 
0.57358E-05 
0.35439E-05 
0.22739E-05 



0. 


10E 


00 


0. 


20E 


00 


0. 


30E 


00 


0. 


4 0E 


00 


0. 


50E 


00 


0. 


60E 


00 


0. 


70E 


00 


0. 


80E 


00 


0. 


90E 


00 


0. 


10E 


01 


0. 


1 IE 


01 


Oo 


12E 


01 


0. 


13E 


Oil 


Oo 


14E 


01 


Oc 


15E 


01 


0. 


16E 


03 


Oo 


17E 


01 


Oo 


18E 


01 


0* 


19E 


01 


Oo 


20E 


01 


Oo 


25E 


01 


= 


30E 


01 


0. 


35E 


01 


Oo 


40E 


01 


= 


45E 


01 


0., 


50E 


01 


0. 


60E 


01 


= 


70E 


1 


Oo 


80E 


oa 


0. 


90E 


0! 


Oo 


10E 


02 


Oo 


12E 


02 


Oo 


14E 


02 


. 


16E 


02 


0„ 


18E 


02 


0« 


20E 


02 


Oo 


22E 


02 


Oo 


24E 


02 


= 


26E 


02 


Oo 


28E 


02 


Oo 


30E 


02 


0. 


35E 


02 


Oo 


40E 


02 


Oo 


45E 


02 


Oo 


50E 


02 


Oo 


60E 


02 


Oo 


70E 


02 


Oo 


80E 


02 


Oo 


90E 


02 


Oo 


10E 


03 



189 



ELECTRIC MICROFIELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTUR3^RS 

A= 0.4000 R= 0.0 TEMP RATIO= 0.25 
CHARGE AT ORIGIN= 9.00 Zl= 1.00 Z2 = 9.00 

E PIE) 

0.10E 00 0.12604E-01 

0.20E 00 0.48683E-01 

0.30E 00 0.10342E 00 

Q.40E 00 0.16990E 00 

O.SOE 00 0.24047E 00 

0.60E 00 0.30798E 00 

0.70E 00 0.36679E 00 

0.80E 00 0.41323E 00 

0.90E 00 0.44564E 00 

0.10E 01 0.46406E 00 

O.HE 01 0.46975E 00 

0.12E 01 0.46471E 00 

0.13E 01 0.45123E 00 

0. 14E 01 0.43160E 00 

0.15E 01 0.40786E 00 

0.16E 01 0.38173E 00 

0.17E 01 0.35458E 00 

0.18E 01 0.32744E 00 

0. 19E 01 0.30105E 00 

0.20E 01 0.27S90E 00 

0.25E 01 0.17495E 00 

0.30E 01 0.11214E 00 

0.3SE 01 0.74449E-01 

0.40E 01 0.51421E-01 

0.45E 01 0.36851E-01 

O.SOE 01 0.27276E-01 

0.60E 01 0.16221E-01 

0.70E 01 0.1Q373E-G1 

0.80E 01 0.70741E-02 

0.90E 01 0.50692E-02 

0.10E 02 0.37794E-02 

O. 12E 02 0.23297E-02 

0.14E 02 0. 16094E-02 

0.16E 02 0.12Z09E-02 

0.18E 02 0.96406E-03 

0.20E 02 0.78928E-03 

0.22E 02 0.64956E-03 

0.24E 02 0.53544E-03 

0.26E 02 0.44217E-03 

0.28E 02 0.36589E-03 

0.30E 02 0.30345E-03 

0.35E 02 0.19222E-03 

0.40E 02 0.12404E-03 

0.45E 02 0.81816E-04 

O.SOE 02 0.55339E-04 

0.60E 02 0.276S3E-04 

0.70E 02 0.1S615E-04 

0.80E 02 0.97681E-05 

0. 90E 02 0.661 89E-05 

0.10E 03 0.47499E-05 



190 



ELECTRIC MICROFIELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBERS 



A= 0.4000 R= 0,0 

CHARGE AT ORlGIN= 9.00 



TEMP RATIOa 0.50 

Zl= 1.00 Z2 = 9.00 



PIE) 



0, 


10E 


00 


0. 


20E 


00 


= 


30E 


00 


Oc 


4 0E 


00 


0. 


50E 


00 


Go 


60E 


00 


0, 


70E 


00 


0. 


80E 


00 


Go 


90E 


00 


0. 


10E 


OS 


Oo 


HE 


01 


Oo 


12E 


01 


Oo 


13E 


01 


Go 


14E 


01 


Oo 


15E 


01 


Oo 


16E 


OS 


0. 


17E 


01 


0. 


18E 


01 


0. 


19E 


01 


Oo 


20E 


01 


0. 


25E 


01 


Oo 


30E 


01 


0. 


35E 


01 


Go 


40E 


01 


0. 


45E 


01 


Oo 


50E 


01 


Qo 


60E 


01 


0. 


70E 


01 


Oo 


80E 


01 


Oo 


90E 


01 


Oo 


10E 


02 


Oo 


12E 


02 


Oo 


14E 


02 


0* 


16E 


02 


Oo 


18E 


02 


0, 


20E 


02 


0. 


22E 


02 


0- 


24E 


02 


0. 


26E 


02 


0. 


28E 


02 


Oo 


30E 


02 


0. 


35E 


02 


Oo 


40E 


02 


0. 


45E 


02 


0, 


50E 


02 


0. 


60E 


02 


0. 


70E 


02 


0. 


80E 


02 


0, 


90E 


02 


0. 


10E 


03 



0.14940E-01 
0.57535E-01 
0.12163E 00 
0. 19851E 00 
0.27869E 00 
0.35359E 00 
0.41671E 00 
0.46418E 00 
0.49463E 00 
0.S0872E 00 
0.50849E 00 
0.49668E 00 
0.47620E 00 
0.44982E 00 
0.41988E 00 
0.38831E 00 
0.35653E 00 
0.32557E 00 
0.29612E 00 
0.26B58E 00 
0.16274E 00 
0.10062E 00 
0.64895E-01 
0.43761E-01 
Q.30726E-01 
0.22339E-01 
0. 12900E-01 
0.80521E-02 
0.53528E-02 
0.37452E-02 
0.27258E-02 
0.15697E-02 
0.97858E-03 
0.64543E-03 
0.441 91 E-03 
0.31229E-03 
0.22728E-03 
0.16999E-03 
0. 13036E-03 
0. 10228E-03 
0.8 1925E-04 
0.50755E-04 
0.34100E-04 
0.24009E-04 
0. 17287E-04 
0.94703E-05 
0.55444E-05 
0.34394E-05 
0.22416E-05 
0.15218E-05 



191 



ELECTRIC MICROFIELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURB-RS 

A= 0-4000 R= 0.0 TEMP RATIO= I. 00 

CHARGE AT ORIGIN^ 9.00 Zl = 1.00 Z 2= 9.00 

E P(E) 

0.19902E-01 
0.762Q7E-01 
0.1596GE 00 
0.25718E 00 
0.35S45E 00 
0.44285E 00 
0.51 146E 00 
0.55742E 00 
0.5805QE 00 
0.58305E 00 
0.56889E 00 
0.54238E 00 
0.50766E 00 
0.46831E 00 
0.42714E 00 
0.38623E 00 
0.34699E 00 
0.31029E 00 
0.27660E 00 
0.24610E 00 
0.13716E 00 
0.79413E-01 
0.48590E-01 
0.31365E-01 
0.2121 1E-01 
0. 14917E-01 
0.80882E-02 
0.4S052E-02 
0.30726E-02 
0.20688E-02 
0. 14353E-02 
0.72804E-03 
0.39389E-03 
0.22635E-03 
0. 13756E-03 
0.88044E-04 
0.59092E-04 
0.41414E-04 
0.30 179E-04 
0.22769E-04 
O. 1771 IE-04 
0, 10502E-04 
0.G8653E-05 
0.46536E-05 
0.32051E-05 
0. 15902E-05 
0.83421E-06 
0.46043E-06 
0.26604E-06 
0.16G13E-06 



0, 


10E 


00 


0. 


20E 


00 


Oo 


30E 


00 


0. 


40E 


00 


0. 


50E 


00 


Oo 


6 0E 


00 


0. 


70E 


00 


= 


80E 


00 


Oo 


9 0E 


00 


0. 


10E 


01 


Oo 


HE 


01 


0. 


12E 


01 


Oo 


13E 


01 


Oo 


14E 


01 


Oo 


15E 


OS 


0. 


16E 


01 


0. 


17E 


01 


Oo 


18E 


3 


Oo 


19E 


01 


0, 


20E 


01 


Oo 


25E 


05 


Do 


30E 


01 


Oo 


35E 


01 


Oo 


40E 


01 


Oo 


45E 


01 


0. 


50E 


01 


0, 


60E 


01 


Oo 


70E 


01 


Oo 


80E 


01. 


Oo 


90E 


01 


0. 


10E 


02 


Go 


12E 


02 


Oo 


14E 


02 


Oo 


16E 


02 


Co 


18E 


02 


Oo 


20E 


02 


Oo 


22E 


02 


0. 


24E 


02 


Oe 


26E 


02 


Oo 


2 8E 


02 


Oo 


30E 


02 


Oo 


35E 


02 


Oo 


40E 


02 


Oo 


45E 


02 


Oo 


50E 


02 


Oo 


60E 


02 


Oo 


70E 


02 


0, 


80E 


02 


Oo 


90E 


02 


Oo 


10E 


03 



192 



ELECTRIC MICROFIELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBERS 



A= 0,4000 R= 0.0 

CHARGE AT ORIGIN= 9.00 



Zl = 



TEMP RATIO= 2,00 

1,00 Z2= 9,00 



P(E) 



0. 10E 00 
0.20E 00 
0.30E 00 
0.40E 00 
0.50E 00 
0.6 0E 00 
0.70E 00 
0.80E 00 
0.90E 00 
0. 10E 01 
0. 1 IE 01 
0- 12E 01 
0.1 3E 01 
0. 14E 01 
0.15E 01 
0. 16E 01 
0. 17E 01 
0. 18E 01 
0. 19E 01 
0.20E 01 
0.25E 01 
0.30E 01 
0.35E 01 
0.40E 01 
0.45E 01 
0.50E 01 
0.60E 01 
0.70E 01 
0.80E 01 
0.90E 01 
0. 10E 02 
0.1 2E 02 
0. 14E 02 
0.16E 02 
0. 18E 02 
0.20E 02 
0.22E 02 
0.24E 02 
0.26E 02 
0.28E 02 
0.30E 02 
0.35E 02 
0.40E 02 
0.45E 02 
0.50E 02 
0.60E 02 
0.70E 02 
0.80E 02 
0.90E 02 
0. 10E 03 



= 


3 1 042E- 


-01 


Oc 


l 1753E 


00 


0, 


24 169E 


00 


Oc 


38000E 


00 


Oc 


50954E 


00 


Oc 


61303E 


00 


0. 


68112E 


00 


0* 


71211E 


00 


Go 


70999E 


00 


0« 


68189E 


00 


Oc 


63588E 


00 


Oc 


57943E 


00 


Oc 


51863E 


00 


0« 


4 5 793 E 


00 


Oc 


40027E 


00 


= 


3 4 735E 


00 


Oc 


29997E 


00 


Oc 


25829E 


00 


Oc 


22209E 


00 


Oc 


19093E 


00 


Oc 


91839E- 


-0 1 


Oc 


47435E- 


-01 


Oc 


26485E- 


-01 


0« 


15835E- 


-0 1 


Oc 


10016E- 


-01 


Oc 


66323E- 


-02 


Oc 


32Q3QE- 


-02 


Oc 


17402E- 


-02 


Oc 


10 238E- 


-02 


Oc 


63476E- 


03 


Oc 


40924E- 


03 


Oc 


18562E- 


-03 


Oc 


92768E- 


-04 


Oc 


50000E- 


-04 


0. 


28454E- 


-04 


Oc 


16877E- 


-04 


Oc 


10402E- 


-04 


Oc 


66439E- 


-05 


Oc 


43869E- 


-05 


0. 


29867E- 


-05 


0. 


20913E- 


•05 


0, 


95390E- 


-06 


= 


49026E- 


-06 


Oc 


27278E- 


-06 


Oc 


15806E- 


-06 


Oo 


5666 7E- 


-07 


Oc 


22000E- 


-07 


Oc 


91 775E- 


-08 


Oc 


40820E- 


-08 


Qc 


19208E- 


-08 



193 



ELECTRIC MICROFIELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBZRS 

A= 0.4000 R= 0.0 TEMP RATIO= 4.00 
CHARGE AT ORIGIN- 9.00 Zl= 1.00 Z2= 9.00 

E P(E) 

0.10E 00 0.57717E-01 

0.20E 00 0.21386E 00 

0.30E 00 0.42460E 00 

0.40E 00 0.63674E 00 

0.50E 00 0.80586E 00 

0.60E 00 0.90717E 00 

0.70E 00 0.93670E 00 

0.80E 00 0.90561E 00 

0.90E 00 0.83223E 00 

0. 10E Oi 0.73545E 00 

0. 1 IE 01 0.63080E 00 

0.12E 01 0.52907E 00 

0.13E 01 0.43659E 00 

0.14E 01 0.35623E 00 

0. 15E 01 0.28856E 00 

0. 16E 01 0.23281E 00 

0.17E 01 0.18756E 00 

0. 18E 01 0.151 18E 00 

0. 19E 01 0. 12212E 00 

0.20E 01 0.98964E-01 

0.25E 01 0.37073E-01 

0.30E 01 0.15799E-01 

0.35E 01 0.75352E-02 

0.40E 01 0.39393E-02 

0.45E 01 0.22067E-02 

0.50E 01 0.13061E-02 

0.60E 01 0.50714E-03 

0.70E 01 0.23346E-03 

0.80E 01 0.11906E-03 

0.90E 01 0.64587E-04 

0.10E 02 0.36683E-04 

0.12E 02 0.13053E-04 

0.14E 02 0.50456E-05 

0. 16E 02 0.20924E-05 

0.18E 02 0.92673E-06 

0.20E 02 0.43647E-06 

0.22E 02 0.21765E-06 

0.24E 02 0.11440E-06 

0.26E 02 0.63103E-07 

0.28E 02 0.36370E-07 

0.30E 02 0.21807E-07 

0.35E 02 0.70106E-08 

0.40E 02 0.26206E-08 

0.45E 02 0.10637E-08 

0.50E 02 0.44795E-09 

0.60E 02 0.87136E-10 



0.70E 02 0.19006E-10 

0.80E 02 0.45989E-11 

0.90E 02 0.12215E-11 

0.10E 03 0.35237E-12 



194 



ELECTRIC MICROFIELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTUR3ERS 



A= 0.4000 
CHARGE AT ORIGIN= 



R= 0.10E 
9.00 Zl = 1.00 



TEMP RATIO= I. 00 
Z2= 9.00 



P(E) 



0.10E 00 

0.20E 00 

0.30E 00 

0.40E 00 

0.50E 00 

0.60E 00 

0.70E 00 

0.80E 00 

O.90E 00 

0.10E 01 

0. 1 IE 01 

0.12E 01 

0. 13E 01 

0.14E 01 

0.1 5E 01 

0. 16E 01 

0. 17E 01 

0. 18E 1 

0. 19E 01 

0.20E 01 

0.25E 01 

0.30E 01 

0.35E 01 

0.40E 01 

0.45E 01 

0.50E 01 

0.60E 01 

0.70E 01 

0.80E 01 

0.90E 01 

0. 10E 02 

0. 12E 02 

0.14E 02 

0- 16E 02 

0. 18E 02 

0.20E 02 

0.22E 02 

0.24E 02 

0.26E 02 

0.28E 02 

0.30E 02 

0.35E 02 

0.40E 02 

0.45E 02 

0.50E 02 

0.60E 02 

0.70E 02 

0.80E 02 

0.90E 02 

O.IOE 03 



0. 19877E-01 
0.75868E-01 
0.1581 IE 00 
0.25319E 00 
0.34751E 00 
0.42996E 00 
0.49347E 00 
0.53515E 00 
0.55555E 00 
0.55744E 00 
0.54463E 00 
0.52113E 00 
0.49058E 00 
0.45600E 00 
0.41972E 00 
0.38342E 00 
0.34827E 00 
0.31501E 00 
0.28405E 00 
0.25559E 00 
0.14939E 00 
0.88827E-01 
0.54741E-01 
0.35097E-01 
0.23368E-01 
0. 16094E-01 
0.83018E-02 
0.47002E-02 
0.28971E-02 
0. 19222E-02 
0. 13574E-02 
0.78017E-03 
0.50242E-03 
0.33512E-03 
0.22710E-03 
0.15630E-03 
G.10919E-03 
0.77396E-04 
0.55638E-04 
0.40545E-04 
0.29938E-04 
0. 14815E-04 
0.78819E-0b 
0.44771E-05 
0.26960E-05 
0. 1 1324E-05 
0.55216E-06 
0.29575E-06 
0. 16850E-06 
0.1 0096E-06 



195 



IN 



ELECTRIC MICROFIELD DISTR IBUTION FUNCT ION 
A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBERS 



A= 0-4000 
CHARGE AT ORIGIN^ 



R= O-lOE 13 
9.00 Z\ 



I. 00 



TEMP RATIO= 
Z2= 



a. oo 

9.00 



P<E) 



0. 10E 00 
0.20E 00 
Q.30E 00 
0.4QE 00 
0.50E 00 
0.60E 00 
0.70E 00 
0.80E 00 
0.90E 00 
0. I0E 01 
0. 1 IE 01 
0. 12E Ot 
0.13E 01 
0.14E 01 
0.1 5E 01 
0. 16E 01 
0. 17E 01 
0.18E 01 
0. 19E 01 
0.20E 01 
0.25E 01 
0.30E 01 
0.35E 01 
0.40E 01 
0.45E 01 
0.50E 01 
0.60E 01 
0.70E 01 
0.80E 01 
0.90E 01 
0. IOE 02 
0.12E 02 
0. 14E 02 
0. 16£ 02 
0. 18E 02 
0.20E 02 
0.22E 02 
0.24E 02 
0.26E 02 
0.28E 02 
0.30E 02 
0.35E 02 
0.40E 02 
0. 45E 02 
0.50E 02 
0.60E 02 
0.70E 02 
0.80E 02 
0.90E 02 
0. IOE 03 



0.20276E-01 
0.77229E-01 
0.16038E 00 
0.25558E 00 
0.34864E 00 
0.42828E 00 
0.48774E 00 
0.52485E 00 
0.54098E 00 
0.S3961E 00 
0.52501E 00 
0.50131E 00 
0.47199E 00 
0.43971E 00 
0.40642E 00 
0.37344E 00 
0.34163E 00 
0.31 150E 00 
0.28335E 00 
0.25729E 00 
0.1S723E 00 
0.96721E-01 
0.60812E-01 
0.39282E-01 
0.26Q80E-01 
0. 17770E-01 
0.88666E-02 
0.47436E-02 
0.27123E-02 
O. 16440E-02 
0. 10476E-02 
0.47865E-03 
0.24755E-03 
0. 14 194E-03 
0.38453E-04 
0.58742E-04 
0.40760E-04 
0.2 89 74E-04 
0.20769E-04 
0. 14972E-04 
0. 10853E-04 
0.49717E-05 
0.23529E-05 
0. 1 1488E-05 
0.57777E-06 
0. 15884E-06 
0.48326E-07 
0. 16083E-07 
0.57877E-08 
0.22259E-08 



196 



ELECTRIC MICROFIELU DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBERS 

A= 0.2000 R= 0.0 TEMP RATIO= 0.25 

CHARGE AT ORIGIN^ 12.00 Z 1 = 12.00 ZZ= 11.00 

E PCE) 

0.10E 00 0.16916E-02 

0.20E 00 0.67025E-02 

0.30E 00 0.14844E-01 

0.40E 00 0.25816E-01 

0.50E 00 0.39221E-01 

0.60E 00 0.54588E-01 

0.70E 00 0.7140GE-01 

Q.80E 00 0.89118E-01 

0.9QE 00 0.10721E 00 

0.10E 01 0.12517E 00 

0. 1 IE 01 0.14253E 00 

0.I2E 01 0.15891E 00 

0. 13E 01 0.1 7397E 00 

0.14E 01 0.18747E 00 

0.15E 01 0.19924E 00 

0. 16E 01 0.20918E 00 

0. 17E 01 0.21726E 00 

0. 18E 01 0.22349E 00 

0.19E 01 0.22794E 00 

0.20E 01 0.23072E 00 

0. 25E 01 0.22446E 00 

0.30E 01 0.19812E 00 

0.35E 01 0.16583E 00 

0.40E 01 0.13508E 00 

0.45E 01 0.10874E 00 

0.50E 01 0.87299E-01 

0.60E 01 0.56923E-01 

0.70E 01 0.38251E-01 

0.80E 01 0.26616E-01 

0.90E 01 0.19149E-01 

0.10E 02 0.14194E-01 

0.12E 02 0.83716E-02 

0.14E 02 0.53603E-02 

0. 16E 02 0.36493E-02 

0.18E 02 0.25885E-02 

0.20E 02 0.18919E-02 

0.22E 02 0.14139E-02 

0.24E 02 0.10879E-02 

0.26E 02 0.84935E-03 

0.28E 02 0.67376E-03 

0.30E 02 0.54257E-03 

0.35E 02 0.33532E-03 

0.40E 02 0.22345E-03 

0.45E 02 0.15843E-03 

0.50E 02 0.11795E-03 

0.60E 02 0.71759E-04 

0.70E 02 0.46224E-04 

0.80E 02 0.30986E-04 

0.90E 02 0.21528E-04 

O. 10E 03 0.15439E-04 



197 



ELECTRIC M1CRQFIELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBERS 

A= 0*2000 R= 0.0 TEMP RATIO= 0.50 

CHARGE AT ORIGIN^ 12.00 Z 1= 12.00 Z2= 11.00 

E P(E) 

0.23951E-02 

0.94670E-02 

0.20884E-01 

0.361 19E-01 

Q.54492E-01 

0.75213E-01 

0.97433E-0 1 

0.12032E 00 

0.14305E 00 

0.16491E 00 

0.18530E 00 

0.20373E 00 

0.219S5E 00 

0.23345E 00 

0.24445E 00 

0.25284E 00 , 

0.25872E 00 

0.26226E 00 

0.26364E 00 

0.26310E 00 

0.23984E 00 

0.20043E 00 

0.16030E 00 

0.12566E 00 

0.97908E-0 1 

0.76431E-01 

0.47616E-01 

0.30872E-01 

0.20858E-01 

0. 14630E-01 

O. 10604E-01 

0.60253E-02 

0.37298E-02 

0.24473E-02 

0. 16820E-02 

0. 12014E-02 

0.88512E-03 

0.66789E-03 

0.51459E-03 

0.40402E-03 

0.32259E-03 

0. 19524E-03 

0.12579E-03 

0.84928E-04 

0.59690E-04 

0.32423E-04 

0. 19155E-04 

0. 1 I875E-04 

0.76647E-05 

0.51272E-05 



Oo 


10E 


00 


= 


2 0E 


00 


Oo 


3 0E 


00 


Oo 


40E 


00 


Oo 


50E 


00 


Oo 


6 0E 


00 


Oo 


70E 


00 


Oc 


80E 


00 


Go 


90E 


00 


Oo 


10E 


01 


= 


1 IE 


OS 


Oo 


12E 


01 


Oo 


13E 


01 


Or, 


14E 


01 


Oo 


I5E 


0! 


Oo 


2 6E 


01 


Oo 


17E 


01 


= 


18E 


01 


Oo 


19E 


01 


Oo 


20E 


oa 


Oo 


25E 


01 


Oo 


30E 


01 


Oo 


35E 


01 


Oo 


40E 


OE 


Oo 


45E 


01 


Oo 


SOE 


01 


0. 


60E 


01 


Oo 


70E 


01 


Oo 


SOE 


01 


0. 


90E 


01 


Oo 


10E 


02 


Oo 


12E 


02 


Oo 


14E 


02 


Oo 


16E 


02 


0. 


18£ 


02 


Oo 


20E 


02 


0. 


22E 


02 


Oo 


24E 


02 


0. 


26E 


02 


Oo 


28E 


02 


0. 


30E 


02 


Oo 


35E 


02 


Oo 


40E 


02 


Oo 


45E 


02 


0. 


SOE 


02 


Oo 


60E 


02 


Oo 


70E 


02 


Oo 


80E 


02 


Oo 


90E 


02 


Oo 


10E 


03 



198 



ELECTRIC MICRQFIELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTUR3ERS 



A= 0.2000 R- 0.0 

CHARGE AT ORIGIN^ 12.00 



TEMP 



Z\~ 12.00 



RATIO= 
Z2= 



1.00 
1 1.00 



P(EI 



0. 10E 00 

0.20E 00 

0.30E 00 

Q.40E 00 

0.50E 00 

0.60E 00 

0.70E 00 

0.80E 00 

0.90E 00 

0. 10E 01 

0. HE 01 

0.12E 01 

0. 13E 01 

0. 14E 01 

0. 15E 01 

0. 16E 01 

0. 17E 01 

0. 18E 01 

0. 19E 01 

0.20E 01 

0.25E 01 

0.30E 01 

0.35E 01 

0.4 0E 01 

0.45E 01 

0.50E 01 

0.60E 01 

0.70E 01 

0.8QE 01 

0.9QE 01 

0. 10E 02 

0.12E 02 

O. 14E 02 

0. 16E 02 

0. 18E 02 

0.20E 02 

0.22E 02 

0.24E 02 

0.26E 02 

0.28E 02 

0.30E 02 

0.35E 02 

0.40E 02 

0.45E 02 

0.50E 02 

0.60E 02 

0. 70E 02 

0.80E 02 

0.90E 02 

0.10E 03 



0.38992E-02 
0. 15350E-01 
0.33636E-01 
0.57642E-0 1 
0.85959E-01 
0.H701E 00 
0.14919E 00 
0.18098E 00 
0.21105E 00 
0.23830E 00 
0.26197E 00 
0.281S5E 00 
0.29684E 00 
0.30783E 00 
0.3 1 4 75E 
0.3 1792E 
0.31775E 00 
0.31 473E 00 
0.30930E 00 
0.30192E 00 
0.24893E 00 
0.19129E 00 
0.14267E 00 
0.10547E 00 
0.78170E-01 
0.58439E-01 
0.33887E-01 
0.20720E-01 
0. 1331 1E-01 
0.89281E-02 
0.62 129E-02 
0.32814E-02 
0. 18969E-02 
O. 1 1696E-02 
0.75767E-03 
0.51224E-03 
0.35912E-03 
0.25944E-03 
0. 19191E-03 
0. 14481E-03 
0. 1 1 130E-03 
0.61897E-04 
0.37462E-04 
0.24145E-04 
0. 16217E-04 
0.77524E-05 
0.39262E-05 
0.20957E-05 
0. I 1729E-05 
0.68471E-06 



199 



IN 



ELECTRIC MICROFIELO DISTRIBUTION FUNCTION 
A PLASMA CONTAINING MULTIPLY CHARGED ION PERTUR3ERS 



A- 0,2000 R= 0.0 

CHARGE AT ORIGIN^ 12.00 



Zl= 12.00 



TEMP RATIO= 2.00 
Z2= 11.00 



PIE) 



0. 10E 00 

0.20E 00 

0.30E 00 

0.40E 00 

0.50E 00 

0.6GE 00 

0.70E 00 

0.8QE 00 

0.90E 00 

0.10E 01 

O. 1 IE 01 

0. 12E 01 

0. 13E 01 

0. 14E 01 

0. 15E 01 

0.16E 01 

0. 17E 01 

0. 18E 01 

0. 19E 01 

0.20E 01 

0.25E 01 

0.30E 01 

0.35E 01 

0.4 0E 01 

0.45E 01 

0.50E 01 

0.60E 01 

0.70E 01 

0.80E 01 

0.90E 01 

0. 10E 02 

0.12E 02 

0. 14E 02 

0.16E 02 

0. 18E 02 

0.2 0E 02 

0.22E 02 

0.24E 02 

0.26E 02 

0.28E 02 

0.30E 02 

0.35E 02 

0.40E 02 

0.45E 02 

0.50E 02 

0.60E 02 

0. 70E 02 

0.80E 02 

0.90E 02 

0.1 OE 03 



0.74 139E-02 
0.28983E-0I 
0.62780E-01 



0.10589E 

0.15479E 

0.20576E 

0.25532E 

0.30053E 

0.33919E 

0.36995E 

0.39223E 

0.40614E 

0.41227E 

0.41 157E 

0-40516E 

0.39419E 

0.37977E 

0.36289E 

0.34441E 

0.32503E 

0.23018E 

0. 15584E 

0.10448E 

0.70517E- 

0.48299E- 



00 
00 
00 
00 
00 
00 
00 
00 
00 
00 
00 
00 
00 
00 
00 
00 
00 
00 
00 
00 
01 
01 



0.33685E-01 
0.17394E-01 
0.95934E-02 
0.56194E-02 
0.34717E-02 
0.22465E-02 
0. 10519E-02 
0.55468E-03 
0.32333E-03 
Q.20481E-03 
0.13858E-03 
0.98458E-04 
0.72203E-04 
0.53740E-04 
0.40232E-04 
Q.30279E-04 
0. 15220E-04 
0.78924E-05 
0.42162E-05 
0.23170E-05 
0.75733E-06 
0.27245E-06 
0.10665E-06 
0.449Q3E-07 
0.20103E-07 



A= 0. 

CHARGE 





200 


ELECTRIC MICROFIELD DISTRIBUTION FUNCTION 


PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBERS 


2000 


R= 0.0 TEMP RATIO= 4.00 


AT ORIGIN^ 


12.00 Zl= 12.00 Z2= 11.00 


E 


PIE) 


0. 10E 


00 0.16406E-01 


0.20E 


00 0.63290E-01 


0.30E 


00 0.13415E 00 


0.40E 


00 0.21 965E 


0.50E 


00 0.30947E 00 


0.60E 


00 0.39400E 00 


0. 70E 


00 0.46576E 00 


0. 80E 


00 0.52004E 00 


0.90E 


00 0.55498E 00 


0, 10E 


01 0.57104E 00 


0. HE 


01 0.57040E 00 


0. 12E 


01 0.55612E 00 


0. I3E 


01 0.53160E 00 


0. 14E 


01 0.50005E 00 


0. I5E 


01 0.46428E 00 


0. 16E 


01 0.42654E 00 


0. 17E 


01 0.38858E 00 


0. ISE 


01 0.35161E 00 


0. 19E 


01 0.3I646E 00 


0. 20E 


01 0.28364E 00 


Q.2SE 


01 0.15851E 00 


0.30E 


01 0.87402E-01 


0.35E 


01 0.49030E-01 


0.40E 


01 0.28312E-01 


0.45E 


01 0.16865E-01 


0.50E 


01 0.10371E-01 


G.60E 


01 0.42660E-02 


0. 70E 


01 0.19341E-02 


0.80E 


01 Q.96267E-03 


0.9QE 


01 0.52187E-03 


0. 10E 


02 0.30568E-03 


0. 12E 


02 0.12816E-03 


0* 14E 


02 0.66566E-04 


0, 16E 


02 0.40190E-04 


0. 18E 


02 0.26467E-04 


0.20E 


02 0.179Q3E-04 


0.22E 


02 0.12164E-04 


0.24E 


02 0.02992E-05 


0. 26E 


02 0.56855E-05 


0.28E 


02 0.39107E-05 


0. 30E 


02 0.27008E-05 


0.35E 


02 0.10892E-05 


0.40E 


02 0.45008E-06 


0.45E 


02 0.19043E-06 


0.50E 


02 0.82452E-07 


0.60E 


02 0.16526E-07 


0. TOE 


02 0.36065E-08 


0.80E 


02 0.85288E~09 


0. 90E 


02 0.21751E-09 


0. 10E 


03 0.59538E-10 



201 



ELECTRIC MICROFIELO DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTUR3ERS 

A= 0.4000 R= 0.0 TEMP RATIO= 0.25 

CHARGE AT ORIGIN= 12.00 21= 12.00 Z2= 11.00 

E PIE) 

0.81058E-02 
0.31499E-01 
0.67568E-01 
0.H246E 00 
0.16171E 00 
0.21090E 00 
0.25623E 00 
0.29489E 00 
G.32524E 00 
0.34671E 00 
0.35966E 00 
0.36501E 00 
0.36405E 00 
0.35814E 00 
0.34858E 00 
0.33648E 00 
0.32274E 00 
0.30807E 00 
0.29 296E 
0.27778E 00 
0.20729E 00 
0.15164E 00 
O.lllOOE 00 
0.82016E-01 
0.61350E-01 
0.46549E-01 
0.28038E-01 
0. 17738E-01 
0. 1 1744E-01 
0.80982E-02 
0.57872E-02 
0.32260E-02 
0.19636E-02 
0. 12822E-02 
0.8871 IE-03 
0.64215E-03 
0.48028E-03 
0.36653E-03 
0.28243E-03 
0.21924E-03 
0. 17142E-03 
0.95520E-04 
0.55463E-04 
0.33465E-04 
0.20926E-04 
0.90064E-0S 
0.43257E-05 
0.22680E-05 
0.12699E-05 
0.74424E-06 



0. 


10E 


00 


0. 


20E 


00 


0. 


3 0E 


00 


0. 


40E 


00 


0. 


50E 


00 


0. 


6 0E 


00 


0. 


70E 


00 


0. 


80E 


00 


0. 


90E 


00 


0. 


10E 


01 


0. 


HE 


01 


0. 


12E 


01 


0. 


13E 


01 


0. 


14E 


01 


0. 


15E 


01 


0. 


16E 


01 


0. 


17E 


01 


0. 


18E 


01 


0. 


19E 


01 


0. 


20E 


01 


0. 


25E 


01 


0. 


30E 


01 


0. 


35E 


01 


0. 


40E 


01 


0. 


45E 


01 


0. 


50E 


01 


0. 


60E 


01 


0. 


70E 


01 


0. 


80E 


01 


0. 


9GE 


01 


0. 


10E 


02 


0. 


12E 


02 


0. 


14E 


02 


0. 


16E 


02 


0. 


18E 


02 


0. 


20E 


02 


0. 


22E 


02 


0. 


24E 


02 


0. 


26E 


02 


0. 


28E 


02 


0. 


30E 


02 


0. 


35E 


02 


0. 


40E 


02 


0. 


45E 


02 


0. 


50E 


02 


0. 


60E 


02 


0. 


70E 


02 


0. 


80E 


02 


0. 


90E 


02 


0. 


10E 


03 



202 



ELECTRIC MICRQFIELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION P£RTURB£R5 



A= 0.4000 R= 0.0 

CHARGE AT ORIGIN- 12.00 



Z 1= 12.00 



TEMP RATIQ= 0.50 
Z2= 11.00 



P(E> 



0. 10E 00 

0.20E 00 

0.3 0E 00 

0.40E 00 

0.50E 00 

0.60E 00 

0.70E 00 

0.80E 00 

0.90E 00 

0. 10E 01 

0. HE 01 

0. 12E 01 

0. 13E 01 

O. 14E 01 

0.15E 01 

0. 16E 01 

0. 17E 01 

0. 18E 01 

0. 19E 1 

0.20E 01 

0.25E 01 

0.30E 01 

0.35E 01 

0.40E 01 

0.45E 01 

0.50E 01 

0.60E 01 

0.70E 1 

0.80E 01 

0.90E 01 

. 1 E 2 

0. 12E 02 

0. 14E 02 

0. 16E 02 

0. 18E 02 

0.20E 02 

0.22E 02 

0.24E 02 

0.26E 02 

0.28E 02 

0.30E 02 

0.35E 02 

0.40E 02 

0.45E 02 

0.50E 02 

0.60E 02 

0. 70E 02 

0.80E 02 

0.9QE 02 

0.1 OE 03 



0. 


14290E- 


-01 


0, 


S4852E- 


-01 


0, 


l 1534E 


00 


0, 


18694E 


00 


Oc 


26028E 


00 


0, 


32730E 


00 


0, 


38235E 


00 


0, 


42258E 


00 


0, 


44761E 


00 


Oc 


45881E 


00 


0-= 


45853E 


00 


0< 


44942E 


00 


0, 


43398E 


00 


0* 


41434E 


00 


0. 


39217E 


00 


0. 


36873E 


00 


0. 


34493E 


00 


0< 


32139E 


00 


0, 


29854E 


00 


= 


27667E 


00 


= 


18554E 


00 


0., 


1 2358E 


00 


0< 


83 192E- 


-01 


0, 


57003E- 


-01 


Oc 


39855E- 


-01 


Oc 


28437E- 


-0 1 


0« 


1 5337E- 


-01 


0, 


88056E- 


-02 


0- 


53475E- 


-02 





34055E- 


-02 


= 


22S53E- 


-02 


Oc 


10858E- 


-02 


0« 


57322E- 


-03 


0. 


32433E- 


-03 


Oc 


19510E- 


-03 


Q< 


12398E- 


-03 


0. 


82695E- 


-04 


0, 


57527E- 


-04 


0, 


4 1471E- 


-04 


= 


30782E- 


-04 


= 


23376E- 


-04 


Oc 


1 2448E- 


-04 


0< 


68373E- 


-05 


0< 


38386E- 


-05 


O a 


22009E- 


-05 


= 


76805E- 


-06 


Oc 


28866E- 


-06 


Qc 


l 1607E- 


-06 


0, 


49599E- 


-07 


0, 


22377E- 


-07 



203 



ELECTRIC MICROFIELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBERS 

A= 0.4000 R= 0.0 TEMP RATIO= I. 00 

CHARGE AT ORIGIN- 12.00 Zl= 12.00 Z 2= 11.00 

E PIE) 

0.29234E-01 
0.10988E 00 
0.22343E 00 
0.34653E 00 
0.45808E 00 
0.54409E 00 
0.S9882E 00 
0.62327E 00 
0.62242E 00 
0.60275E 00 
0.57049E 00 
0.53081E 00 
0.48763E 00 
0.44369E 00 
0.40081E 00 
0.36012E 00 
0.32225E 00 
0.2 8 752 E 
0.25600E 00 
0.22764E 00 
0.12620E 00 
0.71 157E-01 
0.41322E-01 
0.2477SE-01 
0. 15339E-0 1 
0.98159E-02 
0.42972E-02 
0.20876E-02 
0.1 I 110E-02 
0.64174E-03 
0.39861E-03 
0. 18427E-03 
0.10189E-03 
0.62722E-04 
0.40641E-Q4 
0.26794E-04 
0.17770E-04 
0. I 1849E-04 
Q.79436E-05 
0.53535E-05 
0.36268E-05 
0. 1401 1E-05 
0.55828E-06 
0.22922E-06 
0.96858E-07 
0. 18771E-07 
0.40273E-08 
0.94805E-09 
0.24269E-09 
0.66958E-10 



0. 


10E 


00 


0. 


2 0E 


00 


0. 


3 0E 


00 


0. 


40E 


00 


0. 


50E 


00 


0. 


60E 


00 


0. 


70E 


00 


0. 


80E 


00 


0. 


90E 


00 


0. 


10E 


01 


0. 


t IE 


01 


0. 


12E 


01 


0. 


13E 


01 


0. 


14E 


01 


0. 


15E 


01 


0. 


16E 


01 


0. 


17E 


01 


0. 


18E 


01 


0. 


19E 


01 


0. 


20E 


01 


0. 


25E 


01 


0. 


30E 


01 


0. 


35E 


01 


0. 


40E 


01 


0. 


45E 


01 


0. 


50E 


01 


0. 


60E 


01 


0. 


70E 


01 


0. 


80E 


01 


0. 


90E 


01 


0. 


10E 


02 


0. 


12E 


02 


0. 


14E 


02 


0. 


16E 


02 


0. 


18E 


02 


0. 


20E 


02 


0. 


22E 


02 


0. 


24E 


02 


0. 


26E 


02 


0. 


2 8E 


02 


0. 


30E 


02 


0. 


3SE 


02 


0. 


40E 


02 


0. 


45£ 


02 


0. 


50E 


QZ 


0. 


60E 


02 


0. 


70E 


02 


0. 


80E 


02 


0. 


90E 


02 


0. 


10E 


03 



204 



ELECTRIC MICROFIELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTUR3ERS 

A= 0.4000 R= 0.0 TEMP RATI0= 2.00 

CHARGE AT ORIGIN^ 12.00 Zl= 12.00 Z2= 11.00 

E PtEi 

0.10E 00 0.69327E-01 

0.20E 00 0.25100E 00 

0.30E 00 0.48131E 00 

0.40E 00 0.69233E 00 

0.50E 00 0.83912E 00 

0.60E 00 0.907S4E 00 

0.70E 00 0.90655E 00 

O.OOE 00 0.85567E 00 

0.90E 00 0.77537E 00 

0.10E 01 0.68221E 00 

0.1 IE 01 0.58760E 00 

0.12E 01 0.49843E 00 

0. 13E 01 0.41820E 00 

0.14E 01 0.34823E 00 

0. 15E 01 0.28847E 00 

0.16E 01 0.23819E 00 

0.17E 01 0.19632E 00 

0. 18E 01 0.16168E 00 

0.19E 01 0.13317E 00 

0.20E 01 0.10978E 00 

0.25E 01 0.43303E-01 

0.30E 01 0.18016E-01 

0.35E 01 0.78091E-02 

0.40E 01 0.35226E-02 

0.45E 01 0.165I8E-02 

0.50E 01 0.8G419E-03 

0.60E 01 0.21240E-03 

0.70E 01 0.64313E-04 

0.80E 01 0.22120E-04 

0.90E 01 0.85640E-05 

O. 10E 02 0.36980E-05 

0.12E 02 0.92238E-06 

0. 14E 02 0.31901E-06 

0. 16E 02 0.14220E-06 

0.18E 02 0.75921E-07 

0.20E 02 0.45177E-07 

0.22E 02 0.28438E-07 

0.24E 02 0.18294E-07 

0.26E 02 0.1I802E-07 

0.28E 02 0.76281E-08 

0.30E 02 0.49392E-08 

0.35E 02 0.167«35E-08 

0.40E 02 0.57754E-09 

0.45E 02 0.20081E-09 

0.50E 02 0.70593E-10 

0.60E 02 0.90113E-11 

0.70E 02 0.12002E-11 

0.80E 02 0.16665E-12 

0.90E 02 0.24098E-13 

0. 10E 03 0.36259E-14 



205 



ELECTRIC MICRQFIELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBERS 

A= 0*4000 R= 0.0 TEMP RATIO= 4.00 

CHARGE AT ORIGIN^ 12.00 21= 12.00 Z2= 11.00 

E P(E) 

O.IOE 00 0.19266E 00 

0.20E 00 0.64645E 00 

0.30E 00 0.11024E 01 

0.40E 00 0.13677E 01 

0.50E 00 0.14011E 01 

0.60E 00 0.12664E 01 

0.70E 00 0.10489E 01 

0.80E 00 0.81950E 00 

0.90E 00 0.61477E 00 

0. 10E 01 0.44860E 00 

0. HE 01 0.32138E 00 

0.12E 01 0.22758E 00 

0.13E 01 0. 16007E 00 

0. 14E 01 0.1 1224E 00 

0.15E 01 0.78664E-01 

0.16E 01 0.55207E-01 

0.17E 01 0.3Q853E-01 

0. 18E 01 0.27447E-01 

0.19E 01 0.19474E-01 

0.20E 01 0.13882E-01 

0.25E 01 0.23946E-02 

0.30E 01 0.42685E-03 

0.35E 01 0.77142E-04 

0.40E 01 0.14134E-04 

0.45E 01 0.26254E-05 

0.50E 01 0.49435E-06 

0.60E 01 0.18255E-07 

0- 70E 01 0.71143E-09 

0.80E 01 0.29250E-10 

0.90E 01 0.12682E-H 

0.10E 02 0.57962E-13 

0.12E 02 0.14 159E-15 

0.14E 02 0.42504E-18 

0.16E 02 0.15629E-20 

0.18E 02 0.70169E-23 

0.20E 02 0.38344E-25 

0.22E 02 0.25421E-27 

0.24E 02 0.20383E-29 

0.26E 02 0.19702E-31 

0.28E 02 0.22885E-33 

0.30E 02 0.31842E-35 

0.35E 02 0.15687E-39 

0.40E 02 0.22274E-43 

0.45E 02 0.86726E-47 

O.SOE 02 0.88086E-50 

0.60E 02 0.13215E-54 

0.70E 02 0.50470E-58 

0.80E 02 0.32930E-60 

0.90E 02 0.24634E-61 

O.IOE 03 0.14177E-61 



206 



ELECTRIC MICROFIELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTUR3ERS 

A= 0.2000 tt= 0-0 TEMP RATIO= 0.25 

CHARGE AT ORIGIN= 17.00 Zl= I. 00 Z2 = 17.00 

E PIE) 

0.72 126E-02 
0.28232E-01 
Q.61279E-01 
0.10364E 00 
0.15200E 00 
0.20281E 00 
0.25265E 00 
0.29856E 00 
0.33821E 00 
0.37007E 00 
0.39333E 00 
0.40789E 00 
0.41422E 00 
0.413I3E 00 
0.40586E 00 
0.39348E 00 
0.37725E 00 
0.3S832E 00 
0.33 767E 
0.31614E 00 
0.2I413E 00 
0.14010E 00 
0.93015E-01 
0.63797E-01 
0.45362E-01 
0.33359E-01 
0.19766E-01 
0.1267IE-01 
0.86S60E-02 
0.62389E-02 
0.46972E-02 
0.29157E-02 
0. 19356E-02 
0. 13498E-02 
0.98256E-03 
0.74192E-03 
0.57749E-03 
0.46045E-03 
0.37372E-03 
0.30691E-03 
0.2S427E-03 
0. 16473E-03 
0. I I 196E-03 
0. 7943 8E-04 
0.5855QE-04 
0.34995E-04 
0.22981E-04 
0.15976E-04 
0. 11542E-04 
0.86138E-05 



0. 


10E 


00 


= 


2 0E 


00 


0. 


3 0E 


00 


Oo 


40E 


00 


0. 


50E 


00 


0. 


60E 


00 


0. 


7 0E 


00 


0. 


80E 


00 


0-c 


90E 


00 


Oo 


10E 


01 


Oo 


HE 


01 


Oo 


12E 


01 


Oo 


i3E 


01 


0^ 


14E 


01 


Oo 


15E 


oa 


Oo 


I6E 


01 


Oo 


17E 


oa 


Oo 


18E 


01 


Oo 


19E 


01 


Oo 


20E 


01 


Oo 


25E 


01 


Oo 


30E 


01 


Oo 


35E 


01 


0. 


40E 


01 


Oo 


45E 


01 


Oo 


50E 


01 


Oo 


60E 


01 


Oo 


70E 


OJ 


Oo 


80E 


oa 


Oo 


90E 


01 


Oo 


10E 


02 


0. 


12E 


02 


Oo 


14E 


02 


Oo 


16E 


02 


o 


18E 


02 


Oo 


2 0E 


02 


Oo 


22E 


02 


0. 


24E 


02 


Co 


26E 


02 


Oo 


2 8E 


02 


Oo 


30E 


02 


Oo 


35E 


02 


0, 


40E 


02 


Oo 


45E 


02 


Oo 


50E 


02 


Oo 


60E 


02 


Oo 


70E 


02 


Oo 


80E 


02 


Oo 


90E 


02 


Oo 


10E 


03 



207 



ELECTRIC MICROFIELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTJRBzRS 

A= 0.2000 R= 0.0 TEMP RATI0= 0.50 

CHARGE AT ORIGIN= 17.00 Zl= 1.00 Z2= 17.00 

E PIE) 

0.79625E-02 
0.31 132E-01 
0.67453E-01 
0. I 1380E 00 
0.16637E 00 
0.22114E 00 
0.27429E 00 
0.32256E 00 
0.36347E 00 
0.39545E 00 
0.41780E 00 
0.43 057E 00 
0.43445E 00 
0.43052E 00 
0.42010E 00 
0.40459E 00 
0.38536E 00 
0.36365E 00 
0.34Q51E 00 
0.31682E 00 
0.20862E 00 
0.13346E 00 
0.87050E-01 
0.58876E-01 
Q.41384E-01 
0.3O135E-01 
0. 17540E-01 
0.1 I 068E-01 
0.74578E-02 
0.53048E-02 
0.39378E-02 
0.23650E-Q2 
0. 15255E-02 
0. 10261E-02 
0.71607E-03 
0.51 703E-03 
0.38525E-03 
0.29544E-03 
0.23256E-03 
O. 18741E-03 
0.1542QE-03 
0.10191E-03 
0.72354E-04 
0.53095E-04 
0.39725E-04 
0.23457E-04 
0. 14767E-04 
0.98258E-05 
0.68502E-05 
0.49606E-05 



0. 


I OE 


00 


0. 


20E 


00 


0. 


30E 


00 


0. 


4 0E 


00 


0. 


50E 


00 


0. 


60E 


00 


Oo 


70E 


00 


0. 


80E 


00 


0. 


90E 


00 


0. 


I0E 


01 


0. 


I IE 


01 


0. 


12E 


01 


0. 


13E 


01 


0. 


14E 


01 


0. 


15E 


01 


0. 


16E 


01 


Oo 


17E 


01 


0. 


18E 


01 


0. 


19E 


01 


0. 


20E 


01 


0. 


25E 


01 


Oo 


30E 


01 


0. 


35E 


01 


0. 


40E 


01 


0. 


45E 


01 


0. 


50E 


OS 


o 


60E 


01 


0* 


70E 


01 


Go 


80E 


01 


o, 


90E 


01 


0, 


10E 


02 


Oo 


12E 


2 


Oo 


14E 


02 


Oo 


16E 


02 


0. 


18E 


02 


o 


20E 


02 


Oo 


22E 


02 


Oc 


24E 


02 


Oo 


26E 


02 


Oo 


28E 


02 


Oo 


30E 


02 


Oc 


35E 


02 


0. 


40E 


02 


0, 


45E 


02 


Oc 


50E 


02 


Oo 


60E 


02 


Oc 


70E 


02 


= 


80E 


02 


Oo 


90E 


02 


Oc 


10E 


03 



208 



IN 



ELECTRIC MICROFIELD DISTRIBUTION FUNCTION 
A PLASMA CONTAINING MULTIPLY CHARGED ION PERTUR3ERS 



A= 0,2000 R= 0,0 

CHARGE AT ORIGIN= 17,00 



Zl = 



1.00 



TEMP RATIO= I. 00 

Z2= 17,00 



P(£J 



O.IOE 00 

0.20E 00 

0.30E 00 

0.40E 00 

O.SOE 00 

0,60E 00 

0.70E 00 

0.80E 00 

0.90E 00 

0. 10E 01 

0. 1 IE 01 

0. 12E 01 

0. 13E 01 

0.14E 01 

0.15E 01 

0.16E 01 

0,17E 01 

0. 18E 01 

0.19E 01 

0.20E 01 

0.25E 01 

0.30E 01 

0.35E 01 

0.4QE 01 

Q.45E 01 

0.50E 01 

0.60E 01 

0. 70E 01 

0.80E 01 

0.90E 01 

0. 10E 02 

0- 12E 02 

0.14E 02 

0,16E 02 

0.18E 02 

0.20E 02 

0.22E 02 

0.24E 02 

0.26E 02 

0.28E 02 

0.30E 02 

0,35E 02 

0.40E 02 

0.45E 02 

0.50E 02 

0.60E 02 

0.70E 02 

0.80E 02 

0.90E 02 

0, 10E 03 



0.95103E-02 
0.37104E-01 
0.80107E-01 
0.13449E 00 
0.19540E 00 
0.25783E 00 
0.31 71 OE 00 
0.36939E 00 
0.41 198E 00 
0.44331E 00 
0.46293E 00 
0.47134E 00 
0.46970E 00 
0.45958E 00 
0,44276E 00 
0.42100E 00 
0.39593E 00 
0.36898E 00 
0.34129E 00 
0.31 378E 
0.19584E 00 
0.12005E 00 
0.7575GE-01 
0.499GSE-01 
0.34330E-01 
0.24541E-01 
0. 13847E-01 
0.85044E-02 
0.55927E-02 
0.38881E-02 
0.28214E-02 
0. 16142E-02 
0,99775E-03 
0.65892E-03 
0.46090E-03 
0.33851E-03 
0.25877E-03 
0,204l2E-03 
0. 16469E-03 
0. 13473E-03 
0. I 1098E-03 
0.69954E-04 
Q.4553QE-04 
0.3Q542E-04 
0.21079E-04 
0. 10854E-04 
0.61269E-05 
0.37370E-05 
0.24275E-05 
0. 16551E-05 



209 



ELECTRIC MICROFIELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTUR3ERS 

A= 0.2000 R= 0.0 TEMP RATlO= 2.00 

CHARGE AT ORIGIN= 17.00 Zl= 1.00 Z2= 17.00 

E P(E> 

0. 10E 00 

0.20E 00 

0.30E 00 

0.4 0E 00 

0.50E 00 

0.60E 00 

0.70E 00 

0.80E 00 

0.90E 00 

0. 10E 01 

0. 1 IE 01 

0. 12E 01 

0. 13E 01 

0. 14E 01 

0.1 5E 01 

0.16E 01 

0.17E 01 

0.18E 01 

0.19E 01 

0.20E 01 

0.25E 01 

0.30E 01 

0.35E 01 

0.40E 01 

0.45E 01 

0.50E 01 

0.60E 01 

0.70E 01 

0.80E 01 

0.90E 01 

0. 10E 02 

0. 12E 02 

0. 14E 02 

0.16E 02 

0. 18E 02 

0.20E 02 

0.22E 02 

0.24E 02 

0.26E 02 

0.28E 02 

0.30E 02 

0.35E 02 

0.40E 02 

0.45E 02 

0.50E 02 

0.60E 02 

0.70E 02 

0.80E 02 

0.90E 02 

0. 10E 03 



Oo 


12844E- 


-01 


0. 


49898E- 


-01 


0. 


10698E 


00 


0. 


17787E 


00 


0. 


25532E 


00 


0, 


33205E 


00 


Go 


40167E 


00 


= 


45935E 


00 


0* 


5 2 1 I E 


00 


0. 


52880E 


00 


Oc 


53988E 


00 


Oc 


53695E 


00 


= 


52238E 


00 


Oo 


49883E 


00 


Oo 


46895E 


00 


0, 


43516E 


00 


Oo 


39951E 


00 


Oo 


36361E 


00 


Q, 


32865E 


00 


Oo 


29548E 


00 


0* 


16699E 


00 


Oc 


9474 1E- 


-0 1 


0. 


56325E- 


-01 


Oc 


35401E- 


-0 1 


0. 


23423E- 


-01 


0* 


16 194E- 


-01 


Oo 


85 169E- 


-02 


Oo 


50298E- 


-02 


Oo 


32 150E- 


-02 


Oo 


2 1483E- 


-02 


Oo 


14675E- 


-02 


Oo 


72575E- 


-03 


Oo 


38614E- 


-03 


0* 


21987E- 


-03 


Oc 


1332SE 


-0 3 


Oo 


85558E- 


-04 


Oo 


57861E- 


-04 


Oo 


4 1006E- 


-04 


o«. 


30296E- 


-04 


Oo 


2321 1E- 


-04 


Oo 


18345E- 


-04 


Oo 


1 1276E- 


-04 


0. 


750G3E- 


-05 


Oo 


51 121E- 


-05 


Oo 


35383E- 


-05 


0. 


1 7747E- 


-05 


Oo 


94210E- 


-06 


Go 


52652E- 


-06 


Oo 


30817E- 


-06 


Go 


18 789E- 


-06 



210 



ELECTRIC MICROFIELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBERS 

A= 0.2000 R= 0.0 TEMP RATIO= 4.00 

CHARGE AT ORIGIN= 17.00 Zl = 1.00 Z2= 17.00 

E P(E) 

0.20351E-01 
0.78416E-01 
0.16586E 00 
0.27070E 00 
0.37959E 00 
0.48011E 00 
0.56257E 00 
0.62100E 00 
0.65323E 00 
0.66039E 00 
0.64593E 00 
0.61459E 00 
0.57148E 00 
0.52136E 00 
0.46828E 00 
0.41534E 00 
0.36476E 00 
0.31791E 00 
0.27555E 00 
0.23793E 00 
0.11279E 00 
0.55990E-01 
0.30066E-01 
0. 17427E-01 
0.10775E-01 
0.70208E-02 
0.33176E-02 
0.17863E-02 
0.10484E-02 
0.64989E-03 
0.4 1900E-03 
0. 18960E-03 
0.94366E-04 
0.50675E-04 
0.28808E-04 
0. 17131E-04 
0. 10618E-04 
G.68397E-05 
0.45643E-05 
0.31460E-05 
0.2232 8E-05 
0.10556E-05 
0.56051E-06 
0.31872E-06 
0.18675E-06 
0.63498E-07 
0.27253E-07 
0. 1 1663E-07 
0.53242E-08 
0.25710E-0S 



0. 


10E 


00 


0. 


20E 


00 


0. 


30E 


00 


0. 


4 0E 


00 


0. 


5 0E 


00 


0. 


60E 


00 


0. 


70E 


00 


0. 


80E 


00 


0. 


90E 


00 


0. 


10E 


01 


0. 


HE 


01 


0. 


12E 


01 


0. 


13E 


01 


0. 


14E 


01 


0. 


15E 


01 


0. 


16E 


01 


0. 


17E 


01 


Go 


18E 


01 


0. 


19E 


01 


0„ 


20E 


01 


0. 


25E 


01 


0. 


30E 


01 


0. 


35E 


01 


0. 


40E 


01 


o 


4 5E 


01 


0. 


50E 


0! 


0. 


60E 


01 


0. 


70E 


01 


0. 


80E 


01 


0. 


90E 


01 


0. 


10E 


02 


0. 


12E 


02 


0. 


14E 


02 


0. 


16E 


02 


0, 


18E 


02 


0. 


20E 


02 


Oo 


22E 


02 


Oo 


24E 


02 


. 


26E 


02 


Go 


28E 


02 


Oo 


30E 


02 


0. 


35E 


02 


Oo 


40E 


02 


Oo 


45E 


02 


Oo 


50E 


02 


0. 


60E 


02 


Oo 


70E 


02 


Oo 


80E 


02 


Oo 


90E 


02 


Oo 


10E 


03 






211 



ELECTRIC MICROFIELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTUR8=:RS 

A= 0.2000 R~ O.IOE 00 TEMP RATIO= I. 00 

CHARGE AT QRIGIN= 17.00 2 1= 1. 00 12.= 17.00 

E PIE) 

O.IOE 00 0.59614E-02 

0.20E 00 0.23377E-01 

0.30E 00 0.50896E-01 

0.40E 00 0.86448E-01 

0.50E 00 0.12748E 00 

0.60E 00 0.17123E 00 

0.70E 00 0.21499E 00 

0.80E 00 0.25637E 00 

0.90E 00 0.29341E 00 

O.IOE 01 0.32473E 00 

O.llE 01 0.3495QE 00 

0. 12E 01 0.36741E 00 

0. 13E 01 0.37862E 00 

0. 14E 01 0.38360E 00 

0.15E 01 0.38305E 00 

0.16E 01 0.37779E 00 

0.17E 01 0.36869E 00 

0.18E 01 0.35658E 00 

0. 19E 01 0.34225E 00 

0.20E 01 0.32638E 00 

0.25E 01 0.24093E 00 

0.30E 01 0.16801E 00 

0.35E 01 0. 1 1536E 00 

0.40E 01 0.79594E-01 

Q.45E 01 0.55719E-01 

0.50E 01 0.39739E-01 

O.&OE 01 0.21484E-01 

0. 70E 01 0.12495E-01 

0.80E 01 0.77148E-02 

0.90E 01 0.50046E-02 

O.IOE 02 0.33758E-02 

0.12E 02 0.16593E-02 

0.14E 02 0.37726E-03 

0.16E 02 Q.49568E-03 

0.18E 02 0.29791E-03 

0.20E 02 0.18954E-03 

0.22E 02 0.12705E-03 

0.24E 02 0.89298E-04 

0.26E 02 0.65495E-04 

0.28E 02 0.4989QE-04 

0.30E 02 0.39280E-04 

0.35E 02 0.24198E-04 

0.40E 02 0.16510E-04 

0.45E 02 0.11676E-04 

0.50E 02 0.04067E-05 

0.60E 02 0.45832E-05 

0.70E 02 0.26579E-05 

0.80E 02 0.16291E-05 

0.90E 02 0.10486E-05 

O.IOE 03 0.70417E-06 



212 



ELECTRIC MICROFIELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTUR8ERS 



A= 0.2000 

CHARGE AT ORIGlN= 



R= 0.10E 13 
17.00 £1= I. 00 



TEMP RAT10= 1.00 
Z2= 17.00 



P(E) 



o.ioe oo 

0.20E 00 

0.30E 00 

0.40E 00 

0.50E 00 

0.6 0E 00 

0.70E 00 

0.80E 00 

0.90E 00 

0. 10E 01 

0. 1 IE 01 

0. 12E 01 

0.13E 01 

Q.14E 01 

0.15E 01 

0. 16E 01 

0. 17E 01 

0.18E 01 

0. 19E 01 

0.20E 01 

0.25E 01 

0.30E 01 

0.35E 01 

0.40E 01 

0.45E 01 

0.50E 01 

0.60E 01 

0.70E 01 

0.80E 01 

0.90E 01 

0. 10E 02 

0.12E 02 

0.14E 02 

0.16E 02 

0. I8E 02 

0.20E 02 

0.2 2E 02 

0.24E 02 

0.26E 02 

0.28E 02 

0.30E 02 

0.35E 02 

0.40E 02 

0.45E 02 

0.50E 02 

0.60E 02 

0.70E 02 

0.80E 02 

0.90E 02 

0. 10E 03 



0.49992E-02 
0. 19634E-01 
0.42857E-0t 
0.73053E-01 
0.10821E 00 
0.14613E 00 
0.18463E 00 
0.22170E 00 
0.25S69E 00 
0.28535E 00 
0.30986E 00 
0.32882E 00 
0.34221E 00 
0.35028E 00 
0.3S350E 00 
0.35246E 00 
0.34781E 00 
0.34020E 00 
0.33027E 00 
0.31859E 00 
0.24883E 00 
0.18259E 00 
0.13078E 00 
0.93205E-01 
0.66779E-01 
0.48358E-01 
0.26439E-01 
0. 15274E-01 
0.92784E-02 
0.59034E-02 
0.39190E-02 
0. 19316E-02 
0. 10772E-02 
0.65903E-03 
0.43066E-03 
0.29562E-03 
0.20995E-03 
0. 1 5199E-03 
0. 1 1 128E-03 
0.82345E-04 
0.61 559E-04 
0.31022E-04 
0. 16S16E-04 
0.92304E-05 
0.5381 1E-05 
0.20233E-05 
0.83515E-06 
0.36710E-06 
0.17063E-06 
0.83380E-07 



213 



ELECTRIC MICROFIELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURB^RS 

A= 0.4000 R= 0.0 TEMP RATIO= 0,25 

CHARGE AT ORIGIN= 17.00 Z I- 1.00 Z2= 17.00 

E P«E> 

0.10E 00 0.14433E-01 

0.20E 00 0.55630E-01 

0.30E 00 0.11776E 00 

0.40E 00 0.19255E 00 

0.50E 00 0.27092E 00 

0.60E 00 0.34460E 00 

0.70E 00 0.40727E 00 

0.80E 00 0.45503E 00 

0.90E 00 0.48638E 00 

0. 10E 01 0.50180E 00 

0.11E 01 0.50312E 00 

0.I2E 01 0.49290E 00 

0.13E 01 0.47393E 00 

0. 14E 01 0.44888E 00 

0.15E 01 0.42006E 00 

Q.16E 01 0.38938E 00 

0. 17E 01 0.35828E 00 

0.18E 01 0.32781E 00 

0. 19E 01 0.29868E 00 

0.20E 01 0.27134E 00 

0.25E 01 0.16540E 00 

0.30E 01 0.10262E 00 

0.35E 01 0.66329E-01 

0.40E 01 0.44793E-01 

0.45E 01 0.31485E-01 

0.50E 01 0.22912E-01 

0.60E 01 0.13232E-01 

0.70E 01 0.82442E-02 

0.80E 01 0.54982E-02 

0.90E 01 0.38661E-02 

O.IOE 02 0.28337E-02 

0.12E 02 0.16795E-02 

0. 14E 02 0.11015E-02 

0.16E 02 0.78119E-03 

0.18E 02 0.58585E-03 

0.20E 02 0.45440E-03 

0.22E 02 0.35717E-03 

0.24E 02 0.28273E-03 

0.26E 02 0.22534E-03 

0.28E 02 0.18081E-03 

0.30E 02 0.14603E-03 

0.35E 02 0.88025E-04 

0.40E 02 0.55109E-04 

0.45E 02 Q.35755E-04 

0.50E 02 0.23988E-04 

0.60E 02 0.11831E-04 

0.70E 02 0.64959E-05 

0.80E 02 0.39000E-05 

0.90E 02 0.25154E-05 

O.IOE 03 0.17121E-05 



214 



ELECTRIC MICROFIELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBERS 



A= 0,4000 R= 0.0 

CHARGE AT ORIGlN= 17.00 



TEMP RATION 0.50 

Zl= I. 00 Z2 = 17.00 



PIE) 



0-= 


10E 


00 


0. 


20E 


00 


Q. 


30E 


00 


= 


4 0E 


00 


0. 


50E 


00 


0„ 


60E 


00 


0. 


70E 


00 


Q<= 


aoE 


00 


0-o 


90E 


00 


0, 


10E 


01 


0-=, 


1 IE 


01 


0. 


12E 


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0» 


13E 


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0. 


14E 


05 


Oo 


15E 


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0. 


16E 


01 


0. 


17E 


OS 


Oo 


18E 


01 


Qo 


19E 


01 


0, 


20E 


01 


3, 


25E 


01 


Qo 


30E 


01 


0. 


35E 


01 


0. 


40E 


01 


0. 


45E 


01 


Oo 


50E 


01 


Oo 


60E 


01 


0. 


70E 


oa 


Q„ 


80E 


01 


0. 


90E 


01 


0. 


10E 


02 


0. 


12E 


02 


0„ 


14E 


02 


0. 


16E 


02 


Oo 


18E 


02 


0, 


20E 


02 


Oo 


22E 


02 


Oo 


24E 


02 


Oo 


26E 


02 


0. 


28E 


02 


Oo 


30E 


02 


Oo 


35E 


02 


0., 


40E 


02 


Go 


45E 


02 


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02 


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60E 


02 


Oo 


70E 


02 


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80E 


02 


0. 


90E 


02 


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10E 


03 



0.1 8858E-01 
0.72326E-01 
0.15187E 00 
0.24559E 00 
0.34087E 00 
0.42673E 00 
0.49542E 00 
0.54288E 00 
0.56846E 00 
0.57406E 00 
0.56307E 00 
0.53951E 00 
0.50734E 00 
0.47005E 00 
0.43043E 00 
0.39061E 00 
0.35208E 00 
0.31576E 00 
0.2.8222E 00 
0.25168E 00 
0.14142E 00 
0.82232E-01 
0.50441E-01 
0.32616E-01 
0.22089E-01 
0. 15555E-01 
0.83782E-02 
0.50774E-02 
0.33806E-02 
0.24219E-02 
0.18286E-02 
0. I 1468E-02 
0.77776E-03 
0.54961E-03 
0.39371E-03 
0.28477E-03 
0.20793E-03 
0.15323E-03 
O. I 1395E-03 
0.85495E-04 
0.64705E-04 
0.33447E-04 
0. 181 77E-04 
0. 10353E-04 
0.61626E-05 
0.24616E-05 
0.1 1309E-05 
0.58330E-06 
0.32974E-06 
0. 19944E-06 



215 



ELECTRIC MICROFIELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURB5RS 

A= 0.4000 R= 0.0 TEMP RATIO= 1.00 

CHARGE AT ORlGIN= 17.00 Zl= I. 00 Z 2= 17.00 

E P(E) 

0.10E 00 0.28832E-01 

0.20E 00 0.10949E 00 

0.30E 00 0.22624E 00 

0.40E 00 0.35795E 00 

0.50E 00 0.48359E 00 

0.60E 00 0.58669E 00 

0.70E 00 0.65765E 00 

0.80E 00 0.69380E 00 

Q.90E 00 0.69793E 00 

0.10E 01 0.67609E 00 

0. 1 IE 01 0.63561E 00 

0. 12E 01 0.583S6E 00 

0. 13E 01 0.52593E 00 

0.14E 01 0.46726E 00 

0.15E 01 0.41068E 00 

0. 16E 01 0.35812E 00 

0.17E 01 0.31058E 00 

0. 18E 01 0.26840E 00 

0. 19E 01 0.23150E 00 

0.20E 01 0.19954E 00 

0.25E 01 0.96755E-01 

0.30E 01 0.50137E-01 

0.35E 01 0.28042E-01 

0.40E 01 0.16790E-01 

0.45E 01 0.10640E-01 

0.50E 01 0.70631E-02 

0.60E 01 0.34681E-02 

0.70E 01 0.18783E-02 

0.80E 01 0.11058E-02 

0.90E 01 0.69196E-03 

O.IOE 02 0.44993E-03 

0.12E 02 0.20215E-03 

0.14E 02 0.97504E-04 

0. 16c 02 0.50257E-04 

0.18E 02 0.27554E-04 

0.20E 02 0.15994E-04 

0.22E 02 0.97847E-05 

0.24E 02 0.62791E-05 

0.26E 02 0.42074E-05 

0.28E 02 0.29302E-05 

0.30E 02 0.211UE-05 

0.35E 02 0.10468E-05 

0.40E 02 0.57993E-06 

0.45E 02 0.33578E-06 

0.50E 02 0.19872E-06 

0.60E 02 0.74018E-07 

0. 70E 02 0.29749E-07 

0.80E 02 0.12809E-07 

0.90E 02 0.S8654E-08 

O.IOE 03 0.28358E-08 



216 



ELECTRIC MICRQFIELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBERS 

A= 0.4000 R= 0*0 TEMP RATIO= 2.00 

CHARGE AT ORIGIN= 17.00 Zl = I. 00 Z2= 17.00 

E P(E) 



0. 


10E 


00 


0. 


20E 


00 


0, 


30E 


00 


0, 


40E 


00 


Qo 


50E 


00 


0. 


6 0E 


00 


0, 


70E 


00 


0. 


80E 


00 


0. 


90E 


00 


0» 


10E 


0! 


, 


I IE 


01 


Oo 


12E 


01 


0. 


13E 


01 


0. 


14E 


01 


0, 


15E 


01 


Go 


16E 


01 


Oo 


17E 


1 


0. 


18E 


oa 


Oo 


19E 


01 


Oo 


20E 


01 


0. 


25E 


oa 


0. 


30E 


01 


0, 


35E 


01 


Oo 


40E 


01 


0. 


45E 


01 


0, 


50E 


01 


Oo 


60E 


01 


Oo 


70E 


01 


0. 


80E 


01 


0. 


90E 


01 


Oo 


10E 


02 


0, 


i2E 


02 


Oo 


14E 


02 


Oo 


I6E 


02 


Oo 


18E 


02 


o.= 


20E 


02 


0. 


22E 


02 


Oo 


24E 


02 


Oo 


26E 


02 


Oo 


28E 


02 


Oo 


30E 


02 


Oo 


35E 


02 


Oo 


40E 


02 


Oo 


45E 


02 


Oo 


S0E 


02 


Oo 


60E 


02 


Oo 


70E 


02 


Oo 


80E 


02 


Oo 


90 E 


02 


Oo 


10E 


03 



0» 


53124E- 


-01 


0. 


I9777E 


00 


Oo 


39560E 


00 


Oo 


59906E 


00 


Oo 


76687E 


00 


Oo 


S7410E 


00 


Oo 


91431E 


00 


Oo 


89S44E 


00 


Oo 


83319E 


00 


Oo 


74495E 


00 


Oo 


64582E 


00 


Oo 


54687E 


00 


Oo 


45508E 


00 


Oo 


37402E 


00 


Oo 


30 483E 


00 


Oo 


24719E 


00 


Oo 


19998E 


00 


0., 


16174E 


00 


Oo 


13100E 


00 


0. 


10638E 


00 


Oo 


40Q75E- 


-01 


Oo 


171 19E- 


-01 


Oo 


81968E- 


-02 


Oo 


43029E- 


-02 


Oo 


24286E- 


-02 


Oo 


14484E- 


-02 


Oo 


5 7625E- 


03 


Oo 


27136E- 


-0 3 


Oo 


I4313E- 


-03 


0. 


82936E- 


-04 


Oo 


5 1839E- 


-0 4 


Oo 


23630E- 


-04 


Oo 


1 1766E- 


-04 


Oo 


59930E- 


-05 


Oo 


31 150E- 


-05 


Oo 


16514E- 


-05 


Oo 


89257E- 


-06 


Oo 


49157E- 


-06 


Oo 


27573E- 


-06 


Oo 


15744E- 


-06 


Oo 


91474E- 


-07 


Oo 


2S310E- 


-07 


Oo 


77220E- 


-08 


0. 


25782E- 


-08 


0. 


93491E- 


-09 


0« 


15276E- 


-09 


Oo 


31 700E- 


-1 


Oo 


786 16E- 


-1 1 


Q a 


21956E- 


-1 1 


Oo 


667Q0E- 


-12 



217 



ELECTRIC MICROFIELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTUR3ERS 

A= 0.4000 R= 0.0 TEMP RATIO= 4.00 

CHARGE AT ORIGIN= 17,00 2 1= 1.00 Z2= 17.00 

E P(E) 

0.11872E 00 
0.42394E 00 
0.79266E 00 
0.10962E 01 
0.12561E 01 
0.12609E 01 
0. 1 1469E 01 
0.96810E 00 
0.77215E 00 
0.590376 00 
0.43780E 00 
0.31793E 00 
0.22789E 00 
0.16225E 00 
0. 1 1533E 00 
0.82156E-01 
0.58834E-01 
0.42446E-01 
0.30898E-01 
0.22717E-01 
0.55399E-02 
0. 14230E-02 
0.39675E-03 
0. 1 1972E-03 
0.38986E-04 
0. 13659E-04 
0.20641E-05 
0.40354E-06 
0.99694E-07 
0.304G1E-07 
0.1 1 177E-07 
0.24099E-08 
0.32795E-09 
0.37555E-09 
0. 18744E-09 
0.94630E-1 
0.48203E-10 
0.24772E-10 
0. 12841E-10 
0.67142E-1 1 
0.3S404E-1 1 
0.74133E-12 
0. 16328E-12 
0.3 7 76 IE- 13 
0.91540E-14 
0.61529E-15 
0.48901E-16 
0.45319E-17 
0.48301E-18 
0.58387E-19 



0. 


10E 


00 


0. 


2 0E 


00 


0. 


3 0E 


00 


0. 


40E 


00 


0. 


50E 


00 


0. 


60E 


00 


0. 


70E 


00 


Oo 


80E 


00 


Oo 


90E 


00 


o 


10E 


01 


0. 


1 IE 


01 


0. 


12E 


01 


Oo 


13E 


01 


0. 


I4E 


0! 


0. 


I5E 


OS 


Oo 


16E 


Oi 


Oo 


17E 


01 


Oo 


18E 


01 


Oo 


19E 


01 


Oo 


20E 


Oi 


Oo 


25E 


Oil 


0. 


30E 


01 


0. 


35E 


01 


0. 


40E 


01 


0. 


45E 


01 


0* 


50E 


01 


0. 


60E 


01 


Oo 


70E 


01 


0, 


80E 


01 


0. 


90E 


01 


Oo 


10E 


02 


Oo 


12E 


02 


Oo 


14E 


02 


Oo 


16E 


02 


Oo 


18E 


02 


Oo 


20E 


02 


Oo 


22E 


02 


0. 


24E 


02 


Oo 


26E 


02 


0, 


28E 


02 


Oo 


30E 


02 


Oo 


35E 


02 


Oo 


40E 


02 


Oo 


4SE 


02 


Oo 


50E 


02 


Oo 


60E 


02 


0. 


70E 


02 


Oo 


80E 


02 


0* 


90E 


02 


Oo 


10E 


03 



218 



ELECTRIC MICROFIELO DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBERS 

A= 0.4000 R= 0.10E 00 TEMP RATIO= 1.00 

CHARGE AT ORIGIN= 17.00 Zl = I. 00 Z2= 17.00 

E PCE) 

0.39815E-01 
0.14844E 00 
Q.29802E 00 
0.45455E 00 
0.58910E 00 
0.68420E 00 
0.73467E 00 
0.74436E 00 
0.72202E 00 
0.67763E 00 
0.62027E 00 
0.55713E 00 
0.49335E 00 
0.43226E 00 
0.37581E 00 
0.32494E 00 
0.27989E 00 
0.24051E 00 
0.20641E 00 
0.17706E 00 
0.83333E-01 
0.41 135E-01 
0.21560E-01 
0. 12001E-01 
0.70657E-02 
0.43744E-02 
0. 17910E-02 
0.83950E-03 
0.44128E-03 
0.25679E-03 
0. 16332E-03 
0.81912E-04 
0.50222E-04 
0.33961E-Q4 
0.23434E-04 
O. 16321E-04 
0. 1 1470E-04 
0.81331E-05 
0.58172E-05 
0.41963E-05 
0.30524E-05 
0. 14269E-05 
0.69995E-06 
0.35925E-06 
0.19238E-06 
0.61962E-07 
0.22861E-07 
0.94457E-08 
0.42729E-08 
0.20688E-08 



Oc 


10E 


00 


0. 


20E 


00 


Oo 


30E 


00 


0. 


40E 


00 


Qo 


50E 


00 


= 


60E 


00 


0. 


70E 


00 


Oo 


80E 


00 


0, 


90£ 


00 


Oo 


10E 


01 


0. 


HE 


01 


a. 


12E 


01 


0. 


13E 


01 


0. 


14E 


01 


Oc 


15E 


01 


Oo 


16E 


01 


Oo 


17E 


1 


0, 


18E 


01 


Oo 


19E 


01 


Oo 


20E 


01 


Oo 


25E 


01 


0. 


30E 


01 


0. 


35E 


01 


0. 


40E 


01 


0. 


45E 


oa 


Qo 


50E 


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0, 


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70E 


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0* 


80E 


01 


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90E 


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0. 


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02 


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14E 


02 


0. 


16E 


02 


0. 


18E 


02 


Oo 


20E 


02 


0. 


22E 


02 


Oo 


24E 


02 


0* 


26E 


02 


Oo 


28E 


02 


0. 


30E 


02 


0. 


35E 


02 


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40E 


02 


Oo 


45E 


02 


0. 


50E 


02 


Oo 


60E 


02 


Oo 


70£ 


02 


0.-= 


80E 


02 


Oo 


90E 


02 


Oo 


10E 


03 



219 



ELECTRIC MICROFIELD DISTRIBUTION FUNCTION 
IN A PLASMA CONTAINING MULTIPLY CHARGED ION PERTURBERS 

A= 0*4000 R = O.IOE 13 TEMP RATIO= 1.00 

CHARGE AT ORIGIN= 17,00 Zl= I. 00 ZZ~ 17,00 

E PIE) 

O.IOE 00 0.49364E-01 

0.20E 00 0.18114E 00 

0.30E 00 0.35497E 00 

0.40E 00 0.52569E 00 

0.50E 00 0.660Q4E 00 

0.60E 00 0.74305E 00 

0.70E 00 0.77522E 00 

0.80E 00 0.76585E 00 

0.9QE 00 0.72713E 00 

0. 10E 01 0.67Q51E 00 

0.11E 01 0,605Q3E 00 

0, 12E 1 0.53735E 00 

0.13E 01 0-47155E 00 

0.14E 01 0.41018E 00 

0. 15E 01 0.35452E 00 

0. 16E 01 0.30S00E 00 

0.17E 01 0.26156E 00 

0. 18E 01 0.22384E 00 

0. 19E 01 0. 19132E 00 

0.20E 01 0.16345E 00 

0,25E 01 0.74938E-01 

0.30E 01 0.35532E-01 

0.35E 01 0.17624E-01 

0.40E 01 0.91662E-02 

0,4SE 01 0.49835E-02 

0.50E 01 0,28247E-02 

0.60E 01 0.86779E-03 

0.70E 01 0.29878E-03 

0.80E 01 0.11074E-03 

0.90E 01 0.44061E-04 

O.IOE 02 0.18763E-04 

0.12E 02 0.41243E-0S 

0.14E 02 0.11493E-05 

0. 16E 02 0.39676E-06 

0.18E 02 0.16581E-06 

0.20E 02 0.81968E-07 

0.22E 02 0.46841E-07 

0.24E 02 0.30236E-Q7 

0.26E 02 0.21S43E-07 

0.28E 02 0.16556E-07 

0.30E 02 0,l341l£-07 

0.3SE 02 0.85816E-08 

0.40E 02 O.S1922E-08 

0.45E 02 0.29108E-08 

0.50E 02 0.15246E-08 

0.60E 02 0.35260E-09 

0. 70E 02 0.68629E-10 

0.80E 02 0.12013E-10 

0.90E 02 0.20205E-11 

O.IOE 03 0.34802E-12 



APPENDIX J 



TABLES OF STARK BROADENED 
LYMAN SERIES LINE PROFILES 



In this appendix we present tables of the final Stark broadened 
line profiles, computed using the numerical procedures discussed in 
Appendix H. We tabulate the blue wing of each profile since the 

asymmetry present is negligible. The electron temperature is expressed 

-3 

in units of electron volts and the electron density is given in cm 

The third line of parameters in each table heading identifies the 
electric microfield distribution function employed to produce the 
profile. (We designate the net radiator charge by XI.) The frequency 
separations (DELTA OMEGA) are expressed in Rydberg units. The middle 
column gives the unnormalized Stark profile in arbitrary units (relative 
intensity) . The right-hand column gives the Doppler-corrected Stark 
profile, normalized to unit intensity. 



220 















221 


LYMAN 


ALPHA 


PROFILE FOR 


HYDROGENIC 


NEON 




ELECTRON TEMPERATURE^ 1019.20 


ELECTRON 


DENSITY= 0.20E 24 


A=0.20 R= 0-0 


TRATIO=0.25 XI= 


9.0 


Zl= 1.0 Z2= 9.0 


DELTA 


STARK 




STARK+ 


OMEGA 


PROFILE 




DOPPLER 


0.0 




0. 169E 


02 




0.622E 


01 


0.735E- 


-02 


0. 12 BE 


02 




0.612E 


01 


0. 14 7E- 


•0 1 


0. 740E 


01 




0.583E 


01 


0.221E- 


-0 I 


0.439E 


01 




0.539E 


01 


0.2 9 4E- 


-01 


0. 283E 


01 




0.484E 


01 


0.368E- 


-01 


0. 198E 


01 




0.422E 


1 


0.4* 1E- 


-01 


0. 14 7E 


01 




0.359E 


01 


0.51 5E- 


-0 1 


0. 1 16E 


01 




0.299E 


01 


0.588E- 


-01 


0.950E 


00 




0.245E 


01 


0.66 2E- 


•0 I 


0.809E 


00 




0. 1 98E 


01 


0.735E- 


-0 1 


0.712E 


00 




0.160E 


01 


0.882E- 


-01 


0.595E 







0.10 7E 


01 


. 1 3E 


00 


0.535E 


00 




0.773E 


00 


. 1 1 8E 


00 


0.506E 


00 




0.626E 


00 


0. 13 2E 


00 


0.492E 


00 




0.554E 


00 


. 1 4 7E 


00 


0.485E 


00 




0.520E 


00 


0.22 IE 


00 


0.448E 


00 




0.456E 


00 


0.294E 


00 


0.359E 


00 




0.369E 


00 


0.366E 


00 


0.259E 


00 




0.270E 


00 


0.441E 


00 


0. 181E 


00 




0. 189E 


00 


0.588E 


00 


0-899E- 


-01 




0.941E- 


-01 


0.735E 


00 


0.498E- 


-01 




0.519E- 


-01 


LYMAN 


ALPHA 


PROFILE FOR 


HYDROGENIC 


NEON 




ELECTRON TEMPERATURE 


:= 1019.20 


ELECTRON 


DENSITY= 0.20E 24 


A=0.2 R= 0.0 


TRATIQ=1.00 XI = 


9.0 


21= 1.0 Z2= 9.0 


del™ 


k 


STARK 




STARK* 
DOPPLER 


OMEGA 


PROFILE 




0„0 




0. 168E 


02 




0.971E 


01 


0.7 3 5E- 


-02 


0. 127E 


02 




0.924E 


01 


0. 147E- 


I 


0.7 39E 


01 




0.800E 


01 


0.22 IE- 


•0 1 


0.439E 


01 




0.636E 


01 


0.294E- 


•01 


0.284E 


01 




0.4 73E 


01 


0.368E- 


01 


0. 199E 


01 




0.337E 


01 


0.4 4 IE - 


-0 I 


0* 149E 


01 




0.238E 


01 


0*51 5E- 


-0 1 


0. 1 18E 


I 




0. 172E 


01 


0.58 8E- 


I 


0.971E 


00 




0.130E 


01 


0.662E- 


•01 


0.833E 


00 




0.104E 


01 


0. 7 3 5E- 


•0 1 


0.738E 


00 




0.872E 


00 


0.882E- 


■01 


0.624E 


00 




0.691E 


00 


0.103E 


00 


0.568E 


00 




0.608E 


00 


0. 1 1 8E 


00 


0.541E 


00 




0.567E 


00 


0. 132E 


00 


0.528E 


00 




0.54 7E 


00 


0. 14 7E 


00 


0.52 IE 


00 




0.536E 


00 
00 


0,22 IE 


00 


0.469E 


00 




0.480E 


0.294E 


00 


0.361E 


00 




0.371E 


00 


0.368E 


00 


0.251E 


CO 




0.259E 


00 


0.441E 


00 


0. 170E 


00 




0.1 75E 


00 


0.58 8E 


00 


0.81 1E- 


-01 




0.836E- 


•01 


0.735E 


00 


0,438E- 


-01 




0.451E- 


-01 



222 



LYMAN ALPHA PROFILE FOR HYDRQGENIC NEON 
ELECTRON TEMPERATURE= 1019.20 ELECTRON DENSITY= 0.20E 24 



A=0„20 R* 0.0 



TRATI0=4.00 XI= 9.0 21= 1.0 Z2= 9.0 



DELTA 


STARK 


STARK+ 


OMEGA 


PROFILE 


OOPPLER 


0.0 


0. 168E 


02 


0.132E 02 


0.735E-02 


0. 127E 


02 


0.117E 02 


0.147E-0 I 


0.741E 


01 


0.849E 01 


0.221E-01 


0.443E 


01 


0.54 8E 1 


0.294E-0 1 


0.289E 


01 


0.350E 01 


0.368E-0 I 


0.205E 


01 


0.237E 01 


0.441E-0 I 


0. 15bE 


01 


0.173E 01 


0.51 5E-0 1 


0. 125E 


01 


0.136E 01 


0.588E-0 I 


0. 105E 


01 


0.1 12E 01 


0.662E-0 1 


0.924E 


00 


0.971E 00 


0.73 5E-0 1 


0.836E 


00 


0.870E 00 


0.882E-0 I 


0.734E 


00 


0.755E 00 


. 1 3E 


0.686E 


00 


0.701E 00 


0.118E 


0.662E 


00 


0.674E 00 


0.132E 00 


0.64 7E 


00 


0.657E 00 


. 1 4 7E 


0.634E 


00 


0.643E 00 


0,22 IE 00 


0.51 IE 


00 


0.519E 00 


0.294E 00 


0.34 IE 


00 


0.34 7E 


0.368E 00 


0.209E 


00 


0.213E 00 


0.441E 00 


0. 128E 


00 


0. 131E 00 


0.58 8E 


0.545E- 


-01 


0.554E-01 


0.73 5E 


0.275E- 


-01 


0.280E-01 



LYMAN ALPHA PROFILE FOR HYDROGENIC NEON 
ELECTRON TEMPERATURE-- 1019.20 ELECTRON OENSITY= 0.20E 24 

A=0.20 Rss 0.10E 00 TRATID=1.00 XI= 9,0 Zl = 1.0 Z2 = 9.0 



DELTA 


STARK 


STARK+ 


OMEGA 


PROFILE 


DOPPLER 


0.0 


0. 169E 


02 


0.980E 01 


0.735E-02 


0. 127E 


02 


0.9 33E 1 


0, 14 7E-0 1 


0.737E 


01 


0.807E 01 


0.22 IE-01 


0.436E 


01 


0.640E 01 


0.29 4E-0 1 


0. 280E 


01 


0.4 75E 01 


0.368E-01 


0. 195E 


01 


0.337E 01 


0.44 IE-01 


0. 144E 


01 


0.236E 01 


0.51 5E-0 1 


0. 1 12E 


01 


0, 169E 1 


0.58 8E-0 1 


0.912E 


00 


0.126E 01 


0.662E-01 


0. 769E 


00 


0.984E 00 


0.735E-01 


0.669E 


00 


0.810E 00 


0.882E-0 1 


0.546E 


00 


0.618E 00 


. ! 3E 


0.481E 


00 


0.525E 00 


, U 8E 


0.448E 


00 


0.479E 00 


0.132E 00 


0.432E 


00 


0.455E 00 


. 1 4 7E 


0.424E 


00 


0.444E 00 


0.221E 00 


0.4 10E 


00 


0.424E 00 


0.294E 00 


0.356E 


00 


0.369E 00 


0.368E 00 


0.2 79E 


00 


0.290E 00 


0.441E 00 


0.208E 


00 


0.216E 


. 58 8E 


0. 1 I IE 


00 


0.1 16E 


0.735E 00 


0.625E- 


-01 


0.650E-01 



223 



LYMAN ALPHA PROFILE FOR HYDROGEN IC NEON 
ELECTRON TEMPERATURE= 1019.20 ELECTRON DENSITY^ Q.20E 24 

A=0.20 R= 0.10E 13 TRATIO=1.00 Xl = 9.0 Zl = 1«0 Z2 = 9.0 



DELTA 


STARK 


STARK+ 


OMEGA 


PROFILE 


DOPPLER 


0.0 




0. 169E 


02 


0.992E 


01 


0.735E- 


-02 


0. 127E 


02 


0.944E 


01 


0.14 7E- 


•01 


0.73 7E 


01 


0.816E 


01 


0.22 1E- 


■0 1 


0.4 35E 


01 


0.64 7E 


01 


0.294E- 


■01 


0.279E 


01 


0.479E 


01 


0.363E- 


01 


0. 193E 


01 


0.339E 


01 


0.44 IE- 


•0 1 


0. 142E 


01 


0.236E 


01 


0.51 5E- 


01 


0. 109E 


01 


0.1 68E 


01 


0.S8 8E- 


■01 


0.881E 


00 


0. 124E 


01 


0.662E- 


01 


0. 734E 


00 


0.959E 


00 


0.735E- 


■01 


0.630E 


00 


0.780E 


00 


0.88 2E- 


•0 1 


0.501E 


00 


0.579E 


00 


. 1 3E 


00 


0.430E 


00 


0.479E 


00 


. 1 1 8E 


00 


0.392E 


00 


0.427E 


00 


. 1 3 2E 


00 


0.372E 


00 


0.399E 


00 


. 1 4 7E 


00 


0.362E 


00 


0.385E 


00 


0.22 IE 


00 


0.357E 


00 


0.374E 


00 


0.294E 


00 


0.332E 


00 


0.349E 


00 


0.368E 


00 


0.282E 


00 


0.296E 


00 


0,44 IE 


00 


0.226E 


00 


0.237E 


00 


0.58 8E 


00 


0. 135E 


00 


0.142E 


00 


. 7 3 5£ 


00 


0.805E- 


-01 


0.847E- 


-01 



LYMAN ALPHA PROFILE FOR HYDROGEN IC NEON 
ELECTRON TEMPERATURE^ 254.80 ELECTRON DENSITY= 0.20E 24 



A=0.40 R* 0,0 



TRATIO-0.25 X 1= 9.0 2 1= 1.0 Z2 = 9.0 



DELTA 


STARK 


STARK.+ 


OMEGA 


PROFILE 


DOPPLER 


0,0 




0. 121E 


02 


0.838E 01 


0.735E- 


-02 


0. 1 04E 


02 


0.806E 01 


0, 14 7E- 


-0 I 


0.742E 


01 


0.719E 01 


0.221E- 


-01 


0.505E 


01 


0.600E 1 


0.294E- 


-01 


0.353E 


01 


0.4 75E 1 


0.36 8E- 


-01 


0.259E 


01 


0.364E 01 


0.441E- 


-01 


0. 199E 


01 


0.277E 01 


0.51 SE- 


-01 


0. 159E 


01 


0.213E 01 


0.588E- 


01 


0. 133E 


01 


0.168E 01 


0.66 2E- 


01 


0. 1 14E 


01 


0.138E 01 


0.735E- 


-01 


0. 100E 


01 


0. 1 17E 1 


0.882E- 


-0 1 


0.830E 


00 


0.916E 00 


0.10 3E 


00 


0.73 IE 


00 


0.783E 00 


0. 1 I 8E 


00 


0.671E 


00 


0.705E 


. 1 3 2E 


00 


0.630E 


00 


0.655E 00 


0. 14 7E 


00 


0.598E 


00 


0.618E 00 


0.221E 


00 


0.464E 


00 


0.477E 00 


0.294E 


00 


0.331E 


00 


0.34 IE 


0.36 6E 


00 


0.226E 


00 


0.233E 00 


0.44 IE 


00 


0. 154E 


00 


0.159E 00 


0.58 8E 


00 


0.770E- 


-01 


0.793E-01 


0.735E 


00 


0.433E- 


-01 


0.446E-0 1 



224 



LYMAN ALPHA PROFILE FOR HYDROGEN tC NEON 
ELECTRON TEMPERATURE^ 254.90 ELECTRON DENSITY: 



0.20E 24 



L=0.»0 


Ra* O.C 




TRATIO=l .0( 


) Xl= 9 


.0 Zl = 1.0 




DELTA 


STARK 


STARK* 




OMEGA 


PROFILE 


DOPPLER 




0.0 




0. 121E 


02 


0.105E 02 




0.73 5E- 


02 


0. 10SE 


02 


0.971E 01 




0.14 7E- 


1 


0.745E 


01 


0.777E 1 




0.22 1E- 


•0 I 


0.510E 


1 


0.569E 01 




0.294E- 


01 


0.359E 


01 


0.407E 01 




0.368E- 


01 


0.266E 


1 


0.297E 01 




0.44 1E- 


■a i 


0.206E 


01 


0.226E 1 




0.515E- 


•01 


0. 168E 


01 


0.180E 01 




0.58 8E- 


•01 


0. 14 IE 


01 


0.150E 01 




Q.662E- 


•01 


0. 123E 


1 


0.129E 01 




0.7 3 5E- 


•01 


0.1 10E 


01 


0.1 I 4E 01 




Q.882E- 


•01 


0.930E 


00 


0.959E 00 




0. 103E 


00 


0.833E 


00 


0.854E 00 




0.1 11 8E 


00 


0.770E 


00 


0.786E 00 




0. 132E 


00 


0.723E 


00 


0.737E 00 




0.147E 


00 


0.682E 


00 


0.694E 00 




0.22 IE 


00 


0.486E 


00 


0.495E 




0.294E 


00 


0.312E 


00 


0.318E 00 




0.368E 


00 


0. 194E 


00 


0.1 97E 




0.44 IE 


00 


0. 123E 


00 


0.12 5E 




0.588E 


00 


0.558E- 


-01 


0.569E-0 1 




0.735E 


00 


0.297E- 


-01 


0.302E-01 



LYMAN ALPHA PROFILE FOR HYDROGEN IC NEON 
ELECTRON TEMPERATURE= 254.80 ELECTRON DENSITY= 0.20E 24 



R= O.C 




TRATIO=4.00 XI= 9 


.0 Zl= 1.0 


DELTA 


STARK 


STARK* 


OMEGA 


PROFILE 


DOPPLER 


0.0 




0. 123E 


02 


0.119E 02 


0.73 5E- 


02 


0. 107E 


02 


0.105E 02 


0.14 7E- 


-0 1 


0.769E 


01 


0.786E 01 


0.22 IE- 


01 


0.534E 


01 


0.554E 01 


0.294E- 


•01 


0.385E 


01 


0.398E 01 


0.368E- 


1 


0.293E 


01 


0,30 IE 01 


0.44 1E- 


-01 


0.234E 


01 


0.240E 01 


0.51 SE- 


•01 


0. 196E 


01 


0.200E 01 


0.588E- 


•0 1 


0.170E 


01 


0.1 73E 1 


0.662E- 


•0 I 


0. 152E 


01 


0.155E 01 


0.735E- 


■0 I 


0. 139E 


01 


0.141E 01 


0.882E- 


•0 1 


0. 121E 


01 


0.123E 01 


0.103E 


00 


0. 109E 


01 


0.110E 01 


0.11 8E 


00 


0.985E 


00 


0.994E 00 


0. 13 2E 


00 


0. 889E 


00 


0.897E 00 


. 1 4 7E 


00 


0.797E 


00 


0.804E 00 


0.22 IE 


00 


0.4 06E 


00 


0.410E 00 


0.29 4E 


00 


0. 193E 


00 


0.195E 00 


0.368E 


00 


0.986E- 


-01 


0.996E-01 


0.44 IE 


00 


0.563E- 


-01 


0.568E-01 


0.588E 


00 


0.243E- 


-01 


0.246E-01 


0.73 5E 


00 


0. 135E- 


-01 


0. 136E-01 



225 



LYMAN ALPHA PROFILE FOR HYOROGENIC NEON 
ELECTRON TEMPERATURE= 254.80 ELECTRON OENSITY= 0.20E 24 

A=0.4Q Rs= 0.10E 00 TRATIO-1.00 Xl = 9.0 Zl- 1.0 Z2= 9.0 



DELTA 


STARK 


STARK+ 


OMEGA 


PROFILE 


DOPPLER 


0.0 




0. 121E 


02 


0.1 05E 2 


0.735E- 


-02 


0. 104E 


02 


0.971E 01 


. 1 4 7E- 


-01 


0.745E 


01 


0.777E 01 


0.22 1E- 


01 


0.509E 


01 


0.569E 01 


0.294E- 


-01 


0.359E 


01 


0.406E 01 


0.368E- 


-01 


0.265E 


01 


0.296E 01 


0.44 1E- 


-01 


0.206E 


1 


0.225E 01 


0.51 5E- 


-01 


0. 167E 


01 


0.179E 1 


0.588E- 


-01 


0. 140E 


01 


0.149E 01 


0.662E- 


•01 


0. 122E 


01 


0.128E 01 


0.73 5E- 


-0 I 


0. 108E 


01 


0.113E 1 


0.882E- 


-01 


0.916E 


00 


0.94 5E 


0.103E 


00 


0. 81 7E 


00 


0.838E 00 


0.118E 


00 


0.753E 


00 


0.769E 00 


0.13 2E 


00 


0.705E 


00 


0.71 9E 


0.14 7E 


00 


0.664E 


00 


0.677E 00 


0.221E 


00 


0.479E 


00 


0.488E 00 


0.294E 


00 


0.31 7E 


00 


0.323E 00 


0.368E 


00 


0.203E 


00 


0.207E 00 


0.441E 


00 


0. 131E 


00 


0.1 34E 


0.58 8E 


00 


0.599E- 


-01 


0.610E-01 


0.73 5E 


00 


0.312E- 


-01 


0.318E-01 



LYMAN ALPHA PROFILE FOR HYOROGENIC NEON 
ELECTRON TEMPERATURE= 254.80 ELECTRON DENSITY= 0.20E 24 

A=0.40 R- 0.10E 13 TRATIO=1.00 Xl= 9.0 Zl= 1.0 ZZ~ 9.0 



DELTA 


STARK 


STARK+ 


OMEGA 


PROFILE 


DOPPLER 


0.0 


. 1 2 1 E 


02 


0.105E 02 


0.73 5E-0 2 


0. 104E 


02 


0.970E 01 


0.147E-01 


0.744E 


01 


0.776E 01 


0.221E-01 


0.509E 


01 


0.569E 01 


0.294E-0 I 


0.358E 


01 


0.406E 01 


0.368E-QI 


0.265E 


01 


0.296E 1 


0.44 1E-0 1 


0.205E 


01 


0.225E 01 


0.51 5E-01 


0. 166E 


01 


0.179E 1 


Q.58 8E-01 


0.140E 


01 


0.1 49E 1 


0.662E-0 1 


0.121E 


1 


0.128E 01 


0.735E-01 


0. 10 8E 


01 


0.1 13E 1 


0.882E-01 


0.910E 


00 


0.939E 00 


0.10 3E 


0.809E 


00 


0.831E 00 


0.118E 


0.743E 


00 


0.760E 00 


0.13 2E 


0.694E 


00 


0.708E 00 


. 1 4 7E 


0.652E 


00 


0.665E 00 


0.221E 00 


0.470E 


00 


0.4 79E 


0.294E 00 


0.31 7E 


00 


0.323E 00 


0.368E 00 


0.209E 


00 


0.213E 00 


0.441E 00 


0. 138E 


00 


0. 141E 00 


0.588E 00 


0.642E- 


-01 


0.655E-01 


0.735E 00 


0.332E- 


-01 


0.338E-01 



226 



LYMAN BETA PROFILE FOR HYDROGENIC NEON 
ELECTRON TEMPERATUR£= 809.10 ELECTRON OENSITY= 0.10E 24 



A=0-20 R= 0.0 



TRATIQ=0.25 XI = 9,0 Zl= 1-0 Z2= 9.0 



DELTA 


STARK 


STARK+ 


OMEGA 


PROFILE 


DQPPLER 


0.0 


0.243E 


00 


0.358E 


00 


0.735E-02 


0.246E 


00 


0.360E 


00 


0.147E-01 


0.254E 


00 


0.36 7E 


00 


0.221E-01 


0.266E 


00 


0.377E 


00 


0.294E-01 


0.282E 


00 


0.392E 


00 


0.368E-01 


0.302E 


00 


0.410E 


00 


0.44 IE-01 


0.325E 


00 


0.432E 


00 


0.51 5E- 1 


0.350E 


00 


0.456E 


00 


0.588E-01 


0.378E 


00 


0.484E 


00 


0.662E-01 


0.407E 


00 


0.513E 


00 


0.7J5E-01 


0.4 3 7E 


00 


0.544E 


00 


0.882E-0 1 


0.501E 


00 


0.6UE 


00 


0.103E 00 


0.566E 


00 


0.679E 


00 


0.1 18E 


0.630E 


00 


0.74 7E 


00 


0.I32E 00 


0.689E 


00 


0.812E 


00 


. 1 4 7E 


0.743E 


00 


0.871E 


00 


0.221E 00 


0.886E 


00 


0.104E 


01 


0.294E 00 


0.845E 


00 


0.101E 


01 


0.368E 00 


0.721E 


00 


0.869E 


00 


0.441E 00 


0.589E 


00 


0.7 13E 


00 


0.5S8E 


0.383E 


00 


0.465E 


00 


0.73 5E 


0.252E 


00 


0.306E 


00 



LYMAN BETA PROFILE FOR HYDROGENIC NEON 
ELECTRON TEMPERATURE= 809.10 ELECTRON DENSITY^ 0.10E 24 



A=0.20 R= 0.0 



TRATI0=1.00 XI= 9.0 Zl = 1.0 Z2 = 9.0 



del™ 




STARK 


STARK+ 


OMEGA 




PROFILE 


DOPPLER 


0.0 




0.26 7E 


00 


0.335E 00 


0.73 5E- 


02 


0.269E 


00 


0.338E 00 


Q.147E- 


01 


0.278E 


00 


0.34 7E 


0.221E- 


•01 


0.292E 


00 


0.361E 00 


0.29 4E- 


01 


0.310E 


00 


0.381E 00 


0.368E- 


■0 1 


0.332E 


00 


0.405E 00 


0.44 ȣ- 


•01 


0.357E 


00 


0.433E 00 


0.51 SE~ 


■01 


0.386E 


00 


0.465E 00 


0.598E- 


01 


0.4 I 6E 


00 


0.499E 00 


0.662E- 


■0 1 


0.448E 


00 


0.535E 00 


0.735E- 


■01 


0.482E 


00 


0.574E 00 


0.882E- 


■01 


0.552E 


00 


0.653E 00 


. 1 3£ 


00 


0.622E 


00 


0.733E 00 


0.1 18E 


00 


0.689E 


00 


0.810E 00 


0.13 2E 


00 


0. 751E 


00 


0.881E 00 


. 1 4 7E 


00 


0.805E 


00 


0.944E 00 


0.22 IE 


00 


0.933E 


00 


O.llOE 01 


0.2 94E 


00 


0.864E 


00 


0. 102E 01 


0.36 8E 


00 


0.720E 


00 


0.851E 00 


. 4 4 1 E 


00 


0.577E 


00 


0.633E 00 


0.59 8E 


00 


0.365E 


00 


0.432E 00 


0.735E 


00 


0.235E 


00 


0.278E 00 



227 



LYMAN BETA PROFILE FOR HYDROGENIC NEON 
ELECTRON TEMPERATURE= 809.10 ELECTRON OENSITY= 0.10E 24 



A=0.20 R~ 0.0 



TRATIO=4.00 XI = 9.0 Zl = 1.0 Z2= 9.0 



DELTA 


STARK 


STARK+ 


OMEGA 


PROFILE 


DOPPLER 


0.0 


0.350E 


00 


0.401E 


00 


0.735E-02 


0.354E 


00 


0.406E 


00 


0.14 7E-0 1 


0.366E 


00 


0.419E 


00 


0.22 1E-0 1 


0.386E 


00 


0.440E 


00 


0.294E-01 


0. 41 IE 


00 


0.468E 


00 


0.36 8E-0 I 


0.442E 


00 


0.501E 


00 


0.441E-01 


0.477E 


00 


0.540E 


00 


0.51 5E-Q1 


0.516E 


00 


0.583E 


00 


0.588E-0 I 


0.55 7E 


00 


0.628E 


00 


0.662E-01 


0. 600E 


00 


0.676E 


00 


0.735E-0 1 


0.644E 


00 


0.725E 


00 


0.882E-0 1 


0.732E 


00 


0.823E 


00 


0.103E 00 


0.81 7E 


00 


0.918E 


00 


0.1 I 8E 


0.894E 


00 


0.100E 


01 


0.13 2E 


0.959E 


00 


0.108E 


01 


0.14 7E 


0. 101E 


01 


0.1 13E 


01 


0.221E 00 


0.106E 


01 


0. 1 19E 


01 


0.294E 00 


0. 895E 


00 


0.101E 


01 


0.36 8E 


0.694E 


00 


0.781E 


00 


0.441E 00 


0.527E 


00 


0.593E 


00 


0.588E 00 


0.306E 


00 


0.344E 


00 


0.735E 00 


0. I83E 


00 


0.206E 


00 



LYMAN BETA PROFILE FOR HYDROGEN IC NEON 
ELECTRON TEMPERATURE^ 809.10 ELECTRON DENSITY= 0.10E 24 

A=0.20 R= O.IOE 00 TRATIQ=1.00 XI = 9.0 Zl= 1.0 Z2= 9.0 



DELTA 


STARK 


STARK+ 


OMEGA 


PROFILE 


DOPPLER 


0.0 


0.204E 


00 


0.269E 00 


0.735E-02 


0.206E 


00 


0.272E 00 


0, 147E-01 


0.212E 


00 


0.278E 00 


0.22 1E-01 


0.221E 


00 


0.289E 00 


0.294E-0 1 


0.234E 


00 


0.304E 00 


G.368E-0! 


0.250E 


00 


0.322E 00 


0.44 1E-0 1 


0.268E 


00 


0.344E 00 


0.51 5E-01 


0.289E 


00 


0.368E 00 


0.588E-01 


0.31 IE 


00 


0.394E 00 


0.662E-0 1 


0.335E 


00 


0.423E 00 


C.735E-01 


0.360E 


00 


0.453E 00 


0.882E-0 I 


0.413E 


00 


0.51 8E 


0.103E 00 


0.469E 


00 


0.585E 00 


0. 1 1 8E 


0.524E 


00 


0.652E 00 


0.132E 00 


0.5 78E 


00 


0.718E 00 


. 1 4 7E 


0.628E 


00 


0.779E 00 


0.22 IE 


0.793E 


00 


0.983E 00 


0.29 4E 


0.805E 


00 


0.100E 01 


0.368E 


0.724E 


00 


0.902E 00 


0.441E 00 


0.615E 


00 


0.768E 00 


0.58 8E 


0.421E 


00 


0.525E 00 


0.735E 00 


0.286E 


00 


0.357E 00 



228 



LYMAN BETA PROFILE FOR HYDROGENIC NEON 
ELECTRON TEMPERATURE= 809.10 ELECTRON DENSITY= 0.10E 24 



A=0.20 



H- 



0.10E 13 TRATlO=l.00 Xl= 9.0 Zl = 1.0 Z2 = 9.0 



DELTA 


STARK 


STARK*- 


OMEGA 


PROFILE 


DOPPLER 


0.0 


0. 16 7E 


00 


0.231E 00 


0.73SE-02 


0. 168E 


00 


0.233E 00 


0. 14 7E-0 1 


0. 173E 


00 


0.238E 00 


0.22 1E-01 


0. 180E 


00 


0.247E 00 


0.294E-0 I 


0. 190E 


00 


0.259E 00 


0.368E-0 1 


0.202E 


00 


0.274E 00 


0.44 1E-0 1 


0.216E 


00 


0.291E 00 


0.515E-01 


0.232E 


00 


0.311E 00 


0.588E-01 


0.249E 


00 


0.333E 00 


0.662E-Q1 


0.268E 


00 


0.357E 00 


0.735E-0 I 


0.288E 


00 


0.382E 00 


0.832E-01 


0.331E 


00 


0.437E 00 


0.10 3E 


0.376E 


00 


0.495E 00 


0. 1 18E 


0.422E 


00 


0.554E 00 


0.132E 00 


0.468E 


00 


0.613E 


0.147E 00 


0.513E 


00 


0.671E 00 


0.221E 00 


0.683E 


00 


0.892E 00 


0.29 4E 


0.736E 


00 


0.963E 00 


0.36 8E 


0.701E 


00 


0.919E 00 


0.441E 00 


0.625E 


00 


0.820E 00 


0.53 8E 


0.458E 


00 


0.601E 00 


0.735E 00 


0.326E 


00 


0.429E 00 



LYMAN BETA PROFILE FOR HYDROGENIC NEON 
ELECTRON TEMPERATURE^ 202.30 ELECTRON DENSITY= 0. 10E 24 



A=0.40 R= 0.0 



TRATIO=0.25 XI= 9.0 Z 1= 1.0 Z2 = 9.0 



DELTA 


STARK 


STARK+ 


OMEGA 


PROFILE 


DOPPLER 


0.0 


0.382E 


00 


0.471E 


00 


0.73SE-02 


0.384E 


00 


0.474E 


00 


0. 147E-0 t 


0.39 IE 


00 


0.482E 


00 


0.22 1E-0 1 


0.402E 


00 


0.494E 


00 


0.294E-01 


0.418E 


00 


0.51 IE 


00 


0.368E-0 1 


0.438E 


00 


0.532E 


00 


0.44 1E-01 


0.460E 


00 


0.557E 


00 


0.5 1 5E-0 1 


0.485E 


00 


0.584E 


00 


0.58 8E-0 1 


0.51 IE 


00 


0.614E 


00 


0.66 2E-01 


0.539E 


00 


0.645E 


00 


0.735E-0 I 


0.566E 


00 


0.678E 


00 


0.882E-01 


0.626E 


00 


0.744E 


00 


0.10 3E 


0.682E 


00 


0.809E 


00 


. 1 1 8E 


0.735E 


00 


0.869E 


00 


0.13 2E 


0.781E 


00 


0.923E 


00 


. 1 4 7E 


0.819E 


00 


0.968E 


00 


0.221E 00 


0.890E 


00 


. 1 5E 


1 


0. 294E 


0.81 IE 


00 


0.963E 


00 


0.368E 00 


0.679E 


00 


0.808E 


00 


0.44 IE 


0.550E 


00 


0.656E 


00 


0.58 8E 


0.355E 


00 


0.424E 


00 


0.73 5E 


0.234E 


00 


0.2 79E 


00 



229 



LYMAN BETA PROFILE FOR HYDROGEN 1C NEON 
ELECTRON TEMPERATURE= 202.30 ELECTRON DENSITY= 0.10E 24 



A=0.40 R- 0-0 



TRATIO=1.00 XI = 9.0 Zl~ 1.0 Z2 = 9.0 



DELT/> 




STARK 


STARK* 


OMEG/s 


i 


PROFILE 


DOPPLER 


0.0 




0.473E 


00 


0.545E 


00 


0.73 5E- 


•02 


0.475E 


00 


0.548E 


00 


0. 14 7E- 


01 


0.484E 


00 


0.559E 


00 


0.221E- 


Oi 


0.499E 


00 


0.575E 


00 


0.29 4E- 


I 


0.520E 


00 


0.598E 


00 


0.368E- 


•01 


0.544E 


00 


0.625E 


00 


0.44 1E- 


•0 I 


0.572E 


00 


0.656E 


00 


0.51 5E- 


■01 


0.603E 


00 


0.691E 


00 


0.S8 8E- 


•01 


0.636E 


00 


0.72 7E 


00 


0.662E- 


■0 I 


0.670E 


00 


0.765E 


00 


0.735E- 


01 


0.704E 


00 


0.804E 


00 


0.882E- 


■01 


0.771E 


00 


0.880E 


00 


0.10 3E 


00 


0.834E 


00 


0.951E 


00 


. 1 1 8E 


00 


0.889E 


00 


0.101E 


01 


0.132E 


00 


0.934E 


00 


0. 106E 


01 


0.14 7E 


00 


0.969E 


00 


0.1 10E 


01 


0.221E 


00 


0.981E 


00 


0.U2E 


01 


0.29*E 


00 


0.836E 


00 


0.955E 


00 


0.368E 


00 


0.663E 


00 


0.757E 


00 


0.44 IE 


00 


0.514E 


00 


0.587E 


00 


0.SS8E 


00 


0.309E 


00 


0.354E 


00 


0.73 5E 


00 


0. 193E 


00 


0.221E 


00 



LYMAN BETA PROFILE FOR HYOROGENIC NEON 
ELECTRON TEMPERATURE^ 202.30 ELECTRON DENSITY= 0.10E 2* 



A=0.40 R= 0.0 



TRATIO=4.00 XI= 9.0 Zl = 1-0 Z2 = 9.0 



DELTA 


STARK 


STARK+ 


OMEGA 


PROFILE 


DOPPLER 


0.0 


0. 758E 


00 


0.824E 


00 


0.735E-02 


0.763E 


00 


0.829E 


00 


0. 147E-0 I 


0.778E 


00 


0.846E 


00 


0.221E-01 


0.803E 


00 


0.873E 


00 


0.294E-0 1 


0.836E 


00 


0.908E 


00 


0.368E-0 I 


0.876E 


00 


0.950E 


00 


0.44 1E-0 1 


0.919E 


00 


0.997E 


00 


0.51SE-0 1 


0.965E 


00 


0.1 OSE 


01 


0.58 8E-0 1 


0. 101E 


01 


0. 1 10E 


01 


0.662E-01 


0. 106E 


01 


0. 1 14E 


01 


0.735E-01 


0. HOE 


01 


0.1 19E 


02 


0.882E-0 I 


0. 1 18E 


01 


0. 128E 


1 


0.10 3E 


0. 124E 


01 


0.134E 


1 


• 1 I 8E 


0. 127E 


01 


0.138E 


01 


0.132E 


0.129E 


01 


0. 140E 


01 


0.147E 00 


0. 129E 


01 


0.139E 


01 


0.22 IE 


0. 107E 


01 


0. 1 16E 


01 


0.294E 00 


0.781E 


00 


0.84 7E 


00 


0.3&8E 00 


0.551E 


00 


0.597E 


00 


C.441E 00 


0.391E 


00 


0.424E 


00 


0.588E 00 


0.208E 


00 


0.225E 


00 


0.73 5E 


0. I22E 


00 


0.132E 


00 



230 



LYMAN BETA PROFILE FOR HYDROGEN IC NEON 
ELECTRON TEMPERATURE^ 202-30 ELECTRON DENSITY= O.iOE 24 

A=0.40 R= O.IOE 00 TRATIO=1.00 XI= 9.0 Zl = 1-0 Z2= 9-0 



DELTA 


STARK 


STARK* 


OMEGA 


PROFILE 


DOPPLER 


0.0 


0.462E 


00 


0.535E 


00 


0.735E-02 


0.464E 


00 


0.538E 


00 


0. 14 7E-0 1 


0.473E 


00 


0.548E 


00 


0.22 1E-0 1 


0.488E 


00 


0.564E 


00 


0.294E-01 


0.507E 


00 


0.586E 


00 


0.368E-0 1 


0.531E 


00 


0.613E 


00 


0.44 1E-0 1 


0-558E 


00 


0.643E 


00 


0.515E-01 


0.588E 


00 


0.677E 


00 


0.588E-01 


0.620E 


00 


0.712E 


00 


0.662E-0 1 


0.652E 


00 


0.749E 


00 


0.735E-01 


0.686E 


00 


0.787E 


00 


0.882E-01 


0.751E 


00 


0.861E 


00 


0.10 3E 


0.81 IE 


00 


0.930E 


00 


0. 1 1 8E 


0. 865E 


00 


0.990E 


00 


0.1J2E 


0-909E 


00 


0.104E 


01 


. 1 4 7E 


0.943E 


00 


0- I08E 


01 


0.221E 00 


0- 963E 


00 


0.1 10E 


01 


0.294E 00 


0.832E 


00 


0.955E 


00 


0.368E 00 


0.668E 


00 


0.767E 


00 


0.441E 00 


0.523E 


00 


0.600E 


00 


0.588E 00 


0.31 8E 


00 


0.366E 


00 


0.73 5E 


0.200E 


00 


0.230E 


00 



LYMAN BETA PROFILE FOR HYDROGEN IC NEON 
ELECTRON TEMPERATURE= 202.30 ELECTRON DENSITY^ 0.10E 24 

A=0.40 Rss 0.10E 13 TRATI0=1.00 XI= 9.0 Zl= 1.0 Z2= 9-0 



DELTA 


STARK 




STARK+ 


OMEGA 


PROFILE 


DOPPLER 


0.0 


0.457E 


00 


0.5 32E 


00 


0.735E-G2 


0.460E 


00 


0.535E 


00 


0.147E-01 


0.468E 


00 


0.545E 


00 


0.221E-01 


0.483E 


00 


0.561E 


00 


0.294E-01 


0.502E 


00 


0.583E 


00 


0.368E-01 


0.52&E 


00 


0.609E 


00 


0.44 IE- 1 


0.552E 


00 


0.639E 


00 


0.51 5E-0 1 


0. 582E 


00 


0.672E 


00 


0.588E-0 I 


0.613E 


00 


0.707E 


00 


0.66 2E-0 1 


0.645E 


00 


0.743E 


00 


0.735E-0 1 


0.677E 


00 


0.780E 


00 


0.88 2E-0 1 


0.740E 


00 


0.852E 


00 


. 1 3E 


0. 799E 


00 


0.920E 


00 


0.1 1 8E 


0.851E 


00 


0.979E 


00 


0.13 2E 


0.893E 


00 


0.103E 


01 


- 1 4 7E 


0.926E 


00 


0.1 06E 


01 


0.221E 00 


0.946E 


00 


0.109E 


01 


0.29 AC 


0.B24E 


00 


0.950E 


00 


0.368E 00 


0.668E 


oo 


0.770E 


00 


0.441E 00 


0.528E 


00 


0.608E 


00 


G.588E 00 


0.325E 


oo 


0.3 75E 


00 


0.735E 00 


0.206E 


00 


0.237E 


00 






231 



LYMAN ALPHA PROFILE FOR HYDRQGENJC ALUMINUM 
ELECTRON TEMPERATURE^ 1019-20 ELECTRON DENSITY= 0.20E 24 



A=0.20 R= 0.0 



TRATI0=0.25 XI=12.0 Zl=12.0 Z2=11.0 



DELTA 


STARK 


STARK+ 


OMEGA 


PROFILE 


DOPPLER 


0*0 


0.295E 


02 


0.507E 01 


0.735E-02 


0. 147E 


02 


0.502E 01 


0.14 7E-0 1 


0.588E 


01 


0.489E 01 


0.221E-01 


0.297E 


01 


0.468E 01 


0.294E-01 


0. 177E 


01 


0.439E 1 


0.368E-01 


0.U9E 


01 


0.406E 01 


0.441E-01 


0.862E 


00 


0.368E 1 


0.515E-01 


0.667E 


00 


0.329E 01 


0.58 8E-0 I 


Q.543E 


00 


0.289E 01 


0.662E-01 


0.461E 


00 


0.251E 01 


0.735E-0I 


0.406E 


00 


0.215E 01 


0.882E-01 


0.344E 


00 


0.1 53E 1 


. 1 3E 


0.318E 


00 


0.106E 1 


0- I 18E 


0.310E 


00 


0.751E 00 


0.132E 00 


0.31 IE 


00 


0.558E 00 


. 1 4 7E 


0.318E 


00 


0.450E 00 


0.221E 00 


0.348E 


00 


0.355E 00 


0.29 4E 


0.3 2 7E 


00 


0.336E 00 


0.368E 00 


0.2 75E 


00 


0.287E 00 


0.441E 00 


0.217E 


00 


0.229E 00 


Q.S8 8E 


0. 127E 


00 


0.136E 00 


0.735E 00 


0.757E- 


-01 


0.808E-01 



LYMAN ALPHA PROFILE FOR HYDROGENIC ALUMINUM 
ELECTRON TEMPERATURE= 1019.20 ELECTRON OENSlTY= 0.20E 24 



A =0.20 R= 0.0 



TRATIO=1.00 XI=12.0 Z1=12.0 Z2=11.0 



DELTA 


STARK 


STARK+ 


OMEGA 


PROFILE 


DOPPLER 


0.0 


0.293E 


02 


0.882E 1 


0.735E-02 


0. 146E 


02 


0.853E 01 


0.14 7E-0 1 


0.587E 


01 


0.774E 01 


0.221E-01 


0.298E 


01 


0.660E 01 


0.2 9 4E- 1 


0. 181E 


0! 


0.531E 01 


0.36 8E-0 1 


0. 123E 


01 


0.405E 01 


0.44 1E-0 1 


0.917E 


00 


0.296E 01 


0.51 5E-0 1 


0.732E 


00 


0.21 IE 01 


0.58 8E-0 1 


0.61 8E 


00 


0.1 50E 1 


0.66 2E-0 1 


0.547E 


00 


0.109E 01 


0.73 5E-0 I 


0.502E 


00 


0.826E 00 


0.882E-0 1 


0.458E 


00 


0.578E 00 


0.10 3E 


0.44 8E 


00 


0.498E 00 


. 1 1 8E 


0.450E 


00 


0.475E 00 


0.132E 00 


0.458E 


00 


0.472E 00 


. 1 4 7E 


0.466E 


00 


0.475E 00 


0.221E 00 


0.445E 


00 


0.452E 00 


0.294E 00 


0.351E 


00 


0.360E 00 


0.368E 00 


0.253E 


00 


0.261E 00 


0.44 IE 


0. 177E 


00 


0.132E 00 


0.588E 00 


0.86 4E- 


-01 


0.892E-01 


0.735E 00 


0.453E- 


-01 


0.467E-0 1 



232 



LYMAN ALPHA PROFILE FOR HYDROGEN IC ALUMINUM 
ELECTRON TEMPERATURE = 1019,20 ELECTRON DENSITY* 0.20E 24 



A=0.2 R= 0.0 



TRATIO=4.00 XI=12.0 Zl=12.0 Z2=ll.O 



DELTA 


STARK 


STARK+ 


OMEGA 


PROFILE 


DOPPLER 


0.0 


0.293E 


02 


0.14 3E 2 


0.735E-02 


0. 147E 


02 


0.129E 02 


0-14 7E- 1 


0.599E 


01 


0.956E 01 


0.221E-01 


0.313E 


01 


0.606E 01 


0.294E-0 1 


0. 199E 


01 


0.357E 01 


0.368E-O1 


0.144E 


01 


0.217E 1 


0.44 1E-01 


0. 1 16E 


01 


Q.149E 01 


0.S15E-0 1 


0. 101E 


01 


0.117E 01 


0.588E-0 I 


0.926E 


00 


0. 101E 01 


0.662E-01 


0.883E 


00 


0.929E 00 


0.735E-0 1 


0.861E 


00 


0.888E 00 


0.882E-01 


0.848E 


00 


0.857E 00 


0.103E 00 


0.844E 


00 


0.844E 00 


0.118E 


0.828E 


00 


0.827E 00 


0.132E 00 


0.797E 


00 


0.796E 00 


. 1 4 7E 


Q.754E 


00 


0.754E 00 


0.221E 00 


0.458E 


00 


0.462E 00 


0.294E 00 


0.236E 


00 


0.239E 00 


0.368E 00 


0. 120E 


00 


0.1 22E 


0.441E 00 


0.642E- 


-01 


0.649E-01 


0.58 8E 


0.220E- 


-01 


0.222E-01 


* 7 3 5E 


0.960E- 


-02 


0.967E-02 



LYMAN ALPHA PROFILE FOR HYDROGENIC ALUMINUM 
ELECTRON T£MPERATURE= 254.80 ELECTRON DENSITY= 0.20E 24 



= 0.40 


R= 0.0 


TRATI0=0.2£ 


i XI=12 


1.0 Z1=12.0 . 




DELTA 


STARK 




STARK+ 




OMEGA 


PROFILE 


DOPPLER 




0.0 


0.21 IE 


02 


0.825E 01 




0.735E-02 


0. 140E 


02 


0.801E 01 




0. 14 7E-0 1 


0.705E 


01 


0.734E 01 




0.22 1E-0 1 


0.392E 


01 


0.637E 01 




0.294E-0 1 


0.248E 


01 


0.525E 01 




0.368E-0 1 


0. 174E 


01 


0.415E 01 




0.44 1E-01 


0. 132E 


01 


0.317E 1 




0.51 5E- I 


0. 10 7E 


01 


0.238E 01 




0.588E-01 


0.914E 


00 


0.179E 01 




0.662E-01 


0.81 IE 


00 


0.137E 1 




0.73 5E-0 1 


0.744E 


00 


0. i 10E 01 




0.882E-0 I 


0.666E 


00 


0.806E 00 




0.10 3E 


0.628E 


00 


0.687E 00 




. 1 I 8E 


0. 604E 


00 


0.632E 00 




0.132E 


0.58 3E 


00 


0.600E 00 




. 1 4 7E 


0.5&2E 


00 


0.574E 00 




0.221E 00 


0.429E 


00 


0.440E 00 




G.294E 


0.30 IE 


00 


0.310E 00 




0.368E 00 


0.207E 


00 


0.214E 00 




0.441E 00 


0. 143E 


00 


0.148E 00 




0.588E 00 


0.721E- 


-01 


0.743E-01 




0.73 5E 


0.397E- 


-01 


0.409E-01 



233 



LYMAN ALPHA PROFILE FOR HYDROGENIC ALUMINUM 
ELECTRON TEMPERATURE^ 254.80 ELECTRON DENSITY= 0.20E 24 



A=0.40 R= 0.0 



TRATIO=1.00 XI=12.0 Z1=12.0 Z2=ll.O 



DELTA 


i 


STARK 


STARK* 


OMEGA 


PROFILE 


DOPPLER 


0.0 




0.21 IE 


02 


0.127E 02 


0.7 3 5E- 


02 


0. 141E 


02 


0.116E 2 


0. 14 7E- 


1 


0.722E 


01 


0.910E 01 


0.22 IE- 


■01 


0.414E 


01 


0.630E 01 


0.29 4E- 


■01 


0.273E 


01 


0.412E 1 


0.368E- 


01 


0.201E 


01 


0.275E 01 


0.44 1E- 


■01 


0.1 62E 


01 


0.199E 1 


0.51 5E- 


•01 


0.139E 


01 


0.1 58E 1 


0.588E- 


■01 


0. 12 5E 


01 


0.1 36E 01 


0.662E- 


•01 


0. 1 16E 


01 


0.122E 01 


0.735E- 


■0 1 


0. 109E 


01 


0.1 13E 01 


0.88 2E- 


■0 1 


0. 101E 


01 


0.102E 01 


0.10 3E 


00 


0.937E 


00 


0.943E 00 


. 1 1 8E 


00 


0.866E 


00 


0.871E 00 


0.132E 


00 


0.792E 


00 


0.797E 00 


. 1 4 7E 


00 


0.717E 


00 


0.722E 00 


0.221E 


00 


0.392E 


00 


0.397E 00 


0.294E 


00 


0.204E 


00 


0.207E 00 


0.368E 


00 


0. 109E 


00 


0.1 1 IE 00 


0.44 1£ 


00 


Q.619E- 


-01 


0.626E-01 


0.588E 


00 


0.240E- 


-01 


0.242E-01 


0.735E 


00 


0. 1 15E- 


-01 


0.116E-01 



LYMAN ALPHA PROFILE FOR HYDROGENIC ALUMINUM 
ELECTRON TEMPERATURE= 254.80 ELECTRON OENSITY= 0.20E 24 



A=0.40 R= 0.0 



TRATIO=4.00 XI=12.0 Zl=12.0 Z2=ll.O 



DELTA 
OMEGA 



0.0 

0.73 

0. 14 

0.22 

0.29 

0.36 

0.44 

0.51 

0.58 

0.66 

0.73 

0.88 

0.10 

0.11 

0.13 

0. 14 

0.22 

0.29 

0.36 

0.44 

0.58 

0.73 



5E-0 2 
7E-0 1 
1E-0 
4E-0 
8E-0 
1E-0 
5E-0 1 
8E-0 1 
2E-0 1 
5E-0 1 
2E-0 1 
3E 



8E 
2E 
7E 
IE 
4E 
8E 
IE 
8E 
5E 



00 
00 
00 
00 
00 
00 
00 
00 
00 



STARK 


PROFILE 


0.219E 


02 


0. 149E 


02 


0.800E 


01 


0.496E 


01 


0.358E 


01 


0.288E 


01 


0.248E 


01 


0.220E 


01 


0. 199E 


01 


0. 180E 


01 


0. 162E 


01 


0. 12 8E 


01 


0.990E 


00 


0.74 7E 


00 


0.558E 


00 


0.416E 


00 


. 1 1 4E 


00 


0. 482E- 


-01 


0.274E- 


•01 


0. 181E- 


-01 


0.986E- 


-02 


0.623E- 


-02 



STARK+ 


DOPPLER 


0.177E 


02 


0.146E 


02 


0.926E 


1 


0.565E 


I 


0.388E 


01 


0.301E 


01 


0.254E 


01 


0.224E 


01 


0.201E 


01 


0. 1 81E 


01 


0.163E 


1 


0. 129E 


1 


0. 100E 


01 


0.756E 


00 


0.565E 


00 


0.421E 


00 


0. 1 16E 


00 


0.485E- 


-01 


0.2 76E- 


-01 


0. 132E- 


-0 1 


0.991E- 


-02 


0.625E- 


-02 









234 



LYMAN BETA PROFILE FOR HYDROGEN IC ALUMINUM 
ELECTRON TEMP£RATURE= 809.10 ELECTRON DENSITY^ 0.10E 24 



A=0.20 


R= 0.0 




TRATI0=0.25 XI=1£ 


!.0 Zl=12.0 




OEL.TA 




STARK 


STARK+ 




MEG/0 


l 


PROFILE 


DOPPLER 




0.0 




0. 113E 


00 


0.254E 00 




0.735E- 


02 


0. 1 15E 


00 


0.256E 00 




0.14 7E- 


01 


0.122E 


00 


0.261E 00 




0.22 1E- 


01 


0.132E 


00 


0-269E 00 




0.294E- 


01 


0. 146E 


00 


0.280E 00 




0.368E- 


01 


0. 162E 


00 


0.294E 00 




0.44 IE- 


•01 


0.181E 


00 


0.311E 00 




0.51 5E - 


01 


0.203E 


00 


0.331E 00 




Q.588E- 


•01 


0.226E 


00 


0.352E 00 




0.66 2E- 


01 


0.25 IE 


00 


0.376E 00 




0.73 5E- 


•0 1 


0.278E 


00 


0.402E 00 




0.882E- 


■01 


0.336E 


00 


0.458E 




0.103E 


00 


0.396E 


00 


0.51 9E 




0.1 I 8£ 


00 


0.458E 


00 


0.581E 00 




0. 13 2E 


00 


0.51 7E 


00 


0.643E 00 




0.14 7E 


00 


0.573E 


00 


0.703E 00 




0.221E 


00 


0.763E 


00 


0.927E 00 




0.294E 


00 


0.792E 


00 


0.985E 00 




0.368E 


00 


0.725E 


00 


0.920E 00 




0.44 IE 


00 


0.626E 


00 


0.804E 00 




0.588E 


00 


0.440E 


00 


0.570E 00 




. 7 3 5E 


00 


0.3 06E 


00 


0.397E 00 



LYMAN BETA PROFILE FOR HYOROGENIC ALUMINUM 
ELECTRON TEMP£RATURE= 809.10 ELECTRON DENSITY= 0. 10E 24 



A=0.20 R= 0.0 



TRATIO^l.OO XI=12.0 Zl=12.0 Z2=ll.O 



DELT/S 




STARK 


STARK* 


OMEG/> 


i 


PROFILE 


DOPPLER 


0.0 




0. 177E 


00 


0.262E 00 


0.735E- 


■02 


0. 181E 


00 


0.266E 00 


0.14 7E- 


1 


0. 194E 


00 


0.276E 00 


0.22 1E- 


01 


0.213E 


00 


0.294E 00 


C.294E- 


•01 


0.239E 


00 


0.318E 


0.36 8E- 


■01 


0.269E 


00 


0.34 7E 


. 4 4 I E- 


•01 


0.303E 


00 


0.382E 00 


0.51 5E- 


■01 


0.34 IE 


00 


0.421E 00 


0.588E- 


■01 


0.382E 


00 


0.463E 00 


0.662E- 


-0 1 


0.425E 


00 


0.508E 00 


0.7 3 5E- 


•0 I 


0.470E 


00 


0.556E 00 


C.882E- 


■0 1 


0.562E 


00 


0.653E 00 


0.10 3E 


00 


0.652E 


00 


0.750E 00 


. 1 1 8E 


00 


0.736E 


00 


0.841E 00 


. 1 3 2E 


00 


0.81 IE 


00 


0.923E 00 


. 1 4 7£ 


00 


0. 8 73E 


00 


0.993E 00 


0.22 IE 


00 


0.990E 


00 


0.1 14E 01 


0.294E 


00 


0.89 IE 


00 


0.103E 01 


0.368E 





0.730E 


00 


0.852E 00 


0.441E 


00 


0.5 79E 


00 


0.677E 00 


0.588E 


00 


0.358E 


00 


0.419E 


0.73 5E 


00 


0.226E 


00 


0.264E 00 



235 



LYMAN BETA PROFILE FOR HYOROGENIC ALUMINUM 
ELECTRON TEMPERATURE^ 809.10 ELECTRON OENSITY= 0.10E 24 



A=0.2 R= 0.0 



TRATIQ=4. 00 XI=12.0 Zl=12.0 Z2=11.0 



DELTA 


STARK 


STARK* 


OMEGA 


PROFILE 


OOPPLER 


0.0 


0.403E 


00 


0.465E 00 


0.735E-02 


0.4 15E 


00 


0.476E 00 


0.14 7E-0 1 


0.452E 


00 


0.509E 00 


0.22 1E-0 1 


0.508E 


00 


0.561E 00 


0.294E-Q1 


0.579E 


00 


0.629E 00 


0.368E-0 1 


0.661E 


00 


0.710E 00 


0.441E-0 1 


0.751E 


00 


0.799E 00 


0.515E-01 


0.846E 


00 


0.893E 00 


0.588E-01 


0.942E 


00 


0.989E 00 


0.662E-0 1 


0. 104E 


01 


0.108E 01 


0.735E-01 


0. II 3E 


01 


0.1 18E 01 


0.882E-0I 


0.129E 


01 


0.134E 01 


0.10 3E 


0. 142E 


01 


0.1 47E 1 


. 1 1 8£ 


0. 150E 


01 


0.1 55E 01 


0.132E 00 


0. 153E 


01 


0.160E 01 


0.14 7E 


0. 153E 


01 


0.160E 1 


0.221E 00 


0. 122E 


01 


0.128E 01 


0.294E 00 


0.833E 


00 


0.878E 00 


0.368E 00 


0.557E 


00 


0.587E 00 


0.441E 00 


0.375E 


00 


0.395E 00 


0»588E 00 


0. 179E 


00 


0.189E 00 


0.735E 00 


0.929E- 


-01 


0.980E-01 



LYMAN SETA PROFILE FOR HYDROGEN IC ALUMINUM 
ELECTRON TEMPERATURE-= 202.30 ELECTRON DENSITY^ 0.10E 24 



A=0.40 R= 0.0 


TRATIO=0.2S 


i XI=12 


!.0 Z1=12.C 


) 




DELTA 


STARK 


5TARK+ 




OMEGA 


PROFILE 


DOPPLER 




0.0 


0.31 9£ 


00 


0.436E 


00 




0.735E-02 


0.324E 


00 


0.440E 


00 




0.14 7E-0 1 


0.340E 


00 


0.453E 


00 




©•22 1E-0 1 


0.365E 


00 


0-474E 


00 




0.294E-0 1 


0.39 7E 


00 


0.503E 


00 




0.36 8E-01 


0.436E 


00 


0.537E 


00 




0.44 IE-0 1 


0.479E 


00 


0.577E 


00 




0.51 5E- 1 


0.S25E 


00 


0.622E 


00 




0.588E-01 


0.573E 


00 


0.669E 


00 




0.662E-01 


0.622E 


00 


0.717E 


00 




0.735E-0 I 


0.671E 


00 


0.767E 


00 




0.832E-Q 1 


0.765E 


00 


0.863E 


00 




CU10 3E 


0. 84 9E 


00 


0.952E 


00 




« 1 I 8E 


0.918E 


00 


0.103E 


01 




. 1 3 2E 


0.971E 


00 


0. 109E 


01 




0.14 7E 


0. I01E 


01 


0. 1 13E 


01 




0.22 IE 


0.994E 


00 


. 1 1 3E 


01 




0.294E 00 


0. 834E 


00 


Q-958E 


00 




0.368E 00 


0.661E 


00 


0.763E 


00 




0.441E 00 


0.516E 


00 


0.595E 


00 




0.58 8E 


0.31SE 


00 


0.364E 


00 




0.73SE 


0.200E 


00 


0.231E 


00 



236 



LYMAN BETA PROFILE FOR HYDROGEN IC ALUMINUM 
ELECTRON TEMPERATURE= 202.30 ELECTRON DENSITY^ 0.10E 24 



A=0.40 R- 0.0 



TRATI0=1.00 XI=12.0 Zl=12.0 Z2=lt.0 



DELTA 
OMEGA 



STARK 
PROFILE 



STARK+ 
DOPPLER 



0.0 

0.73 

0.14 

0.22 

0.29 

0.36 

0.44 

0.51 

0.58 

0.66 

0.73 

0.88 

0.10 

0.1 1 

0.13 

0.14 

0.22 

0.29 

0.36 

0.44 

0.58 

0.73 



5E-0 2 
7E-0 1 
IE-0 1 
4E-0 1 
8E-0 1 
1E-0 I 
5E-0 1 
8E-0 I 
2E-0 1 
5E-0 1 
2E-0 1 
3E 



8E 
2E 
7E 
IE 
4E 
8E 
IE 
8E 
5E 



00 
00 
00 
00 
00 
00 
00 
00 
00 



0.612E 
0.623E 
0.659E 
0.713E 
0.781E 
0.859E 
0.941E 
O. 103E 
0. I 1 IE 
0. 118E 
0. 126E 
0. 137E 
0. 145E 
0. 148E 
0.148E 
0. 146E 
O . 1 I 2E 
0.770E 
0.521E 
0.358E 
O. 180E 



00 
00 
00 
00 
00 
00 
00 
01 
01 
01 
01 
01 
01 
01 
01 
01 
01 
00 
00 
00 
00 



0.998E-0 1 



0.669E 
0.700E 
0.732E 
0.782E 
0.846E 
0.921E 
0.100E 
0.1 09E 
0.1 17E 
0. 125E 
0. 132E 
0. 144E 
0.152E 
0.1 56E 
0. 156E 
0.1 54E 
0.U9E 
0.819E 
0.555E 
0.381E 
0.192E 
0. 106E 



00 
00 
00 
00 
00 
00 
01 
01 
01 
01 
01 
01 
01 
01 
01 
01 
01 
00 
00 
00 
00 
00 



LYMAN BETA PROFILE FOR HYDROGENIC ALUMINUM 
ELECTRON TEMPERATURE^ 202.30 ELECTRON DENSITY= 0.10E 24 



A=0.40 


ft- 0.0 




DELTA 




OMEGA 




0.0 




0.735E-02 




0. 14 7E-0 1 




0.22 1E-01 




0.29 4E-0 I 




0.368E-0 1 




0.44 IE-0 1 




0.51 5E-0 1 




0.58 8E-O1 




0.662E-0 1 




0.735E-0 I 




0.882E-01 




. 1 3E 




0.118E 




0.13 2E 




. 1 4 7E 




0.22 IE 00 




0.2 9 4E 


, 


0.368E 00 




0.441E 00 




0.588E 00 




0.73 5E 



TRATIO=4.00 XI=12.0 21=12.0 22=11.0 



STARK+ 
DOPPLER 

0.153E 01 
0.156E 01 
0.165E 01 
0.177E 01 
0.192E 01 
0.206E 01 
0.219E 01 
0.230E 01 
0.237E 01 
0.241E 01 
0.242E 01 
0.236E 01 
0.221E 01 
0.201E 01 
0.180E 01 
0.159E 01 
0.822E 00 
0.445E 00 
0.262E 00 
0.168E 00 
0.858E-01 
0.526E-01 



STARK 


PROFILE 


0. 145E 


01 


0. 149E 


01 


0. 157E 


01 


0. 170E 


01 


0. 185E 


01 


0. 199E 


01 


0.212E 


01 


0.223E 


01 


0. 230E 


01 


0.234E 


01 


0.235E 


01 


0. 22 8E 


01 


0.213E 


01 


0. 194E 


01 


0. 173E 


01 


0. 15 3E 


01 


0. 792E 


00 


0.423E 


00 


0.252E 


00 


0. 162E 


00 


0.828E- 


-01 


0.507E- 


-01 



237 



LYMAN ALPHA PROFILE FOR HYDROGENIC ARGON 
ELECTRON TEMPERATURE^ 1019.20 ELECTRON DENSITY= 0.20E 24 



A=0.20 


R= 0.0 


TRATIO=0.2S 


> xi=n 


'.0 Zl= 1.0 




DELTA 


STARK 


STARK+ 




OMEGA 


PROFILE 


OOPPLER 




0.0 


0.558E 


02 


0.378E 01 




0.735E-02 


0. 109E 


02 


0.377E 01 




0.147E-01 


0.330E 


01 


0.373E 01 




0.221E-01 


0. 163E 


01 


0.366E 01 




0.294E-0 I 


0. 106E 


01 


0.358E 01 




0.368E-01 


0.820E 


00 


0.347E 01 




0.441E-0 1 


0.723E 


00 


0.335E 01 




0.515E-01 


0.693E 


00 


0.320E 01 




0.58 8E-0 1 


0.697E 


00 


0.305E 01 




0.662E-01 


0.714E 


00 


0.288E 01 




0.735E-01 


0.73SE 


00 


0.271E 01 




0.882E-0 1 


0.765E 


00 


0.236E 01 




0.10 3E 


0.782E 


00 


0.202E 01 




. 11 8E 


0.7 79E 


00 


0.170E 01 




0.132E 00 


0.741E 


00 


0.141E 01 




0.147E 00 


0.677E 


00 


0.1 I 7E 1 




0.22 IE 


0.364E 


00 


0.4 79E 




0.294E 


0. 183E 


00 


0.241E 00 




0.368E 00 


0.100E 


00 


0.130E 00 




0.44 IE 00 


0. 600E- 


-01 


0.738E-0 1 




0.58 8E 


0.266E- 


-01 


0.301E-01 




0.735E 00 


0. 146E- 


-01 


0.157E-01 



LYMAN ALPHA PROFILE FOR HYDROGENIC ARGON 
ELECTRON TEMPERATURE^ 1019.20 ELECTRON DENSITY= 0.20E 24 



A=0.20 R= 0.0 



TRATIO=1.00 XI=17.0 Z 1= 1.0 Z2=17.0 



DELTA 


STARK 


STARK+ 


OMEGA 


PROFILE 


DOPPLER 


0.0 


0.557E 


02 


0.680E 01 


0.735E-02 


0. 109E 


02 


0.671E 01 


0. 14 7E-0 1 


0.333E 


01 


0.643E 01 


0.22 IE-01 


0. 167E 


01 


0.599E 1 


0.294E-0 1 


O.lllE 


01 


0.543E 01 


0.36 8E-0 1 


0.889E 


00 


0.481E 01 


0.44 1E-0 1 


0.807E 


00 


0.416E 1 


0.51SE-01 


0.79 IE 


00 


0.352E 01 


0.58 8E-0 1 


0.805E 


00 


0.294E 01 


0.662E-01 


0.830E 


00 


0.243E 01 


0.73SE-01 


0.853E 


00 


0.201E 01 


0.882E-01 


0.878E 


00 


0.140E 01 


. 1 3E 


0.879E 


00 


0.105E 01 


0.118E 00 


0.853E 


00 


0.868E 00 


0.132E 00 


0.788E 


00 


0.762E 


. 1 4 7E 


0.700E 


00 


0.684E 00 


0.221E 00 


0.330E 


00 


0.359E 00 


0.294E 00 


0. 152E 


00 


0.169E 00 


. 3 6 8E 


0. 787E- 


-01 


0.851E-01 


0.441E 00 


0.453E- 


-01 


0.480E-01 


0.58 8E 


0. 190E- 


-01 


0. 197E-01 


0.73 5E 


0.999E- 


-02 


0. 102E-0 1 



238 



LYMAN ALPHA PROFILE FOR HYOROGENIC ARGON 
ELECTRON TEMPERATURE^ 1019.20 ELECTRON DENSITY= 0.20E 24 

A=0.20 R= 0.0 TRATIQ=4.00 XI=17.0 Zl = 1.0 Z2=17.0 



DELTA 


STARK 


STARK+ 


OMEGA 


PROFILE 


DOPPLER 


0.0 




0.558E 


02 


0.125E 02 


0.735E- 


•02 


0.110E 


02 


0.118E 02 


Q.147E- 


•0 1 


0.346E 


01 


0.999E 1 


0.221E- 


•01 


0. 185E 


01 


0.764E 01 


Q.294E- 


-01 


0.1 34E 


01 


0.537E 01 


0.368E- 


I 


0. 1 17E 


01 


0.359E 01 


0.441E- 


■01 


0. 1 13E 


01 


0.241E 01 


0.5 15E- 


-0 1 


0. 1 16E 


01 


0.174E 01 


0.588E- 


-01 


0. 120E 


01 


0.141E 01 


0.662E- 


-01 


0. 123E 


01 


0.127E 01 


0.735E- 


-01 


0. 123E 


01 


0.121E 01 


0-882E- 


-01 


0. 1 18E 


01 


0. 1 14E 1 


0.10 3E 


00 


0.1 08E 


01 


0.105E 01 


0.11 8E 


00 


0.951E 


00 


0.933E 00 


0.132E 


00 


0.792E 


00 


0.793E 00 


. 1 4 7E 


00 


0.634E 


00 


0.652E 00 


0.22 IE 


00 


0. 196E 


00 


0.206E 00 


C.294E 


00 


0.707E- 


-01 


0.736E-0 1 


C.368E 


00 


0.320E- 


-01 


0.328E-01 


0.441E 


00 


0. 170E- 


-01 


0.174E-01 


.58 8E 


00 


0.666E- 


-02 


0.673E-02 


0.73 5E 


00 


0.346E- 


-02 


0.348E-02 



LYMAN ALPHA PROFILE FOR HYDROGENIC ARGON 
ELECTRON TEMPERATURE= 1019.20 ELECTRON DENSITY^ 0.20E 24 

A-0.20 R= 0.10E 00 TRATIO=1.00 XI=17.0 Z 1= 1.0 Z2=17.0 



DELTyi 


t 


STARK 


STARK+ 


OMEGA 


PROFILE 


DOPPLER 


0.0 




0.557E 


02 


0.671E 01 


0.735E- 


•02 


0. 109E 


02 


0.661E 01 


0.147E- 


-0 I 


0.328E 


01 


0.633E 1 


0.221E- 


■01 


0. 160E 


01 


0.589E 01 


0.29 4E- 


01 


0. 102E 


01 


0.533E 01 


0.368E- 


■0 I 


0.779E 


00 


0.469E 1 


0.44 l£- 


■0 1 


0.673E 


00 


0.403E 01 


0.51 5E- 


-01 


0. 635E 


00 


0.340E 01 


0.58 8E- 


■0 1 


0.631E 


00 


0.281E 01 


0.662E- 


-0 t 


0.643E 


00 


0.230E 1 


0.73 5E- 


■0 1 


0.661E 


00 


0.187E 01 


0.88 2E- 


-0 1 


0.693E 


00 


0.127E 01 


0. 103E 


00 


0.719E 


00 


0.933E 00 


0.1 18E 


00 


0.731E 


00 


0.770E 00 


0.132E 


00 


0.712E 


00 


0.690E 00 


. 1 4 7E 


00 


0.669E 


00 


0.642E 00 


0.221E 


00 


0.4 10E 


00 


0.421E 00 


0.29 4E 


00 


0.222E 


00 


0.236E 00 


0.36 8E 


00 


0. 121E 


00 


0.1 29E 


0.441E 


00 


0.68 7E- 


-01 


0.728E-01 


0.58 8E 


00 


0.263E- 


-01 


0.274E-01 


0.735E 


00 


0. 125E- 


-01 


0. 128E-0 1 



239 



LYMAN ALPHA PROFILE FOR HYDROGENIC ARGON 
ELECTRCN TEMP£RATURE= 1019.20 ELECTRON DENSITY= 0.20E 2* 

A=0.20 R= 0.10E 13 TRATI0=1.Q0 XI=17.0 Zl = 1-0 Z2=17.0 



DELTA 




STARK 


STARK* 


OMEG/ 


i 


PROFILE 


DOPPLER 


0.0 




0.557E 


02 


0.669E 01 


0.735E- 


02 


0. 109E 


02 


0.659E 01 


0.14 7E- 


1 


0.326E 


01 


0.630E 01 


0.22 te- 


01 


0. 158E 


01 


0„586E 01 


0.29 4E- 


•01 


0.997E 


00 


0.530E 01 


0.368E- 


01 


0.74SE 


00 


0.466E 1 


0.44 1E- 


01 


0.632E 


00 


0.400E 01 


0.51 SE - 


■01 


0.587E 


00 


0.336E 01 


0.588E- 


•01 


0.576E 


00 


0.277E 01 


0.66 2E- 


-0 1 


0.583E 


00 


0.225E 01 


0.73 5E- 


01 


. 5 9 7E 


00 


0.182E 01 


0.832E- 


•01 


0..627E 


00 


0.1 22E 1 


0.10 3E 


00 


0.6S5E 


00 


0.885E 00 


0.11 8E 


00 


0.675E 


00 


0.726E 00 


0.132E 


00 


0.667E 


00 


0.652E 00 


. 1 4 7E 


00 


0.63 7E 


00 


0.612E 00 


0.221E 


00 


0.427E 


00 


0.432E 00 


0.294E 


00 


0.248E 


00 


0.26 IE 


0.36 8E 


00 


0. 141E 


00 


0.14 9E 


0.44 IE 


00 


0.822E- 


-01 


0.867E-0 1 


. 58 8E 


00 


0.313E- 


-01 


0.327E-01 


0.735E 


00 


0. 144E- 


-01 


0. 148E-0 1 



LYMAN ALPHA PROFILE FOR HYDROGENIC ARGON 
ELECTRON TEMPERATURE= 254.80 ELECTRON DENSITY= 0.20E 2< 



A=0.40 R= 0.0 



TRATIO=0.25 Xl=17.0 Zl= 1.0 Z2=17.0 



DELTA 


STARK 


STARK+ 


OMEGA 


PROFILE 


DOPPLER 


0«0 


0.4 16E 


02 


0.662E 1 


0.735E-02 


0. 136E 


02 


0.653E 01 


0. 147E-0 1 


0.470E 


01 


0.627E 01 


0.221E-01 


0.244E 


01 


0.587E 01 


0.294E-0 1 


0. 163E 


1 


0.536E 01 


0.368E-0 I 


0. 129E 


01 


0.477E 01 


0.44 IE-01 


0. 1 14E 


01 


0.41 7E 1 


0.51 5E-0 1 


0. 108E 


01 


0.357E 01 


0.588E-01 


0. 105E 


01 


0.302E 01 


0.662E-0 I 


0. 104E 


01 


0.253E 01 


0.73SE-01 


0. 104E 


01 


0.212E 01 


0.882E-01 


0. 101E 


01 


0. 151E 01 


0.10 3E 


0.947E 


00 


0. 1 14E 1 


. 1 1 8E 


0.864E 


00 


0.924E 00 


0.13 2E 


0.76 7E 


00 


0.788E 00 


0.14 7E 


0.668E 


00 


0.688E 00 


0.221E 00 


0.304E 


00 


0.336E 00 


0.294E 00 


0. 145E 


00 


0.160E 00 


0.36 8E 


0.781E- 


-01 


0.839E-01 


0.441E 00 


0.466E- 


-01 


0.491E-0 1 


0.58 8E 


0.205E- 


-01 


0.212E-01 


0.7 3 5E 


0. 1 10E- 


-01 


0.113E-0 1 



240 



LYMAN ALPHA PROFILE FOR HYDROGEN IC ARGON 
ELECTRON TEMPERATURE= 25*. 80 ELECTRON DENSITY= 0.20E 24 



A=0.40 R= 0.0 



TRATIO=1.00 XI=17.0 Zl = 1.0 Z2=17.0 



DELTA 


STARK 


STARK+ 


OMEGA 


PROFILE 


DOPPLER 


0.0 


0.416E 


02 


0.1 I 7E 2 


0.735E-02 


0. 137E 


02 


0.1 HE 02 


0. 147E-0 1 


0.488E 


01 


0.957E 01 


0.22 1E-0 1 


0.266E 


01 


0.751E 01 


0.294E-0 1 


0. 190E 


01 


0.548E 01 


0.368E-01 


0. 161E 


01 


0.385E 01 


0.441E-01 


0. 149E 


01 


0.272E 01 


0.51 5E-01 


0. 145E 


01 


0.204E 01 


0.58 8E-0 1 


0. 14 3E 


01 


0.1 67E 1 


0.662E-01 


0. 140E 


01 


0.147E 01 


0.7 3 5E-0 1 


0. 136E 


01 


0.136E 01 


0.882E-0 1 


0.124E 


01 


0. 121E 1 


0.10 3E 


0. 107E 


01 


0.106E 01 


0. 118E 


0.899E 


00 


0.907E 00 


0.132E 00 


0.732E 


00 


0.751E 00 


. 1 4 7E 


0.586E 


00 


0.610E 00 


0.221E 00 


0.191E 


00 


0.201E 00 


0.294E 00 


0.753E- 


-01 


0.761E-01 


Q.368E 00 


0.363E- 


-01 


0.372E-01 


0.441E 00 


0.203E- 


-01 


0.207E-01 


0.588E 00 


Q.851E- 


-02 


0.860E-02 


0.735E 00 


0.456E- 


-02 


0.459E-02 



LYMAN ALPHA PROFILE FOR HYDROGEN1C ARGON 
ELECTRON TE MPER ATURE= 254.80 ELECTRON DENSITY^ 0.20E 24 



A=0.40 R= 0.0 



TRATIO=4.00 Xl=17.0 Zi= 1.0 Z2=17.0 



DELTA 


STARK 


STARK* 


OMEGA 


PROFILE 


DOPPLER 


0.0 


0.422E 


02 


0.201E 02 


0.735E-02 


0. 143E 


02 


0.1 69E 2 


0.147E-01 


0.5&IE 


01 


0.106E 02 


0,22 IE-0 1 


0.354E 


01 


0.588E 01 


0.294E-01 


0.289E 


01 


0.366E 1 


0.368E-0 I 


0.263E 


01 


0.283E 1 


0.44 IE-01 


0.246E 


01 


0.249E 01 


0.51 5E-01 


0.228E 


01 


0.226E 01 


0.58 8E-0 1 


0.206E 


01 


0.204E 01 


0.662E-01 


0. 180E 


01 


0.180E 1 


0.735E-0 1 


0. 154E 


01 


0.1 55E 1 


0.88 2E-0 1 


0, 105E 


01 


0.1 08E 1 


. 1 3E 


0.690E 


00 


0.719E 00 


0.118E 


0.445E 


00 


0.46 7E 


0.132E 00 


0.291E 


00 


0.305E 00 


0.147E 00 


0. 195E 


00 


0.204E 00 


0.221E 00 


0.473E- 


-01 


0.480E-01 


0.294E 00 


0.212E- 


-01 


0.214E-01 


0.368E 00 


0. 127E- 


-0 I 


0.127E-0 I 


0.441E 00 


Q.856E- 


-02 


0.859E-02 


0.588E 00 


0.471E- 


-02 


0.473E-02 


0.73 5E 


0.299E- 


-02 


0.300E-02 



241 



LYMAN ALPHA PROFILE FOR HYDROGEN IC ARGON 
ELECTRON TEMPERATURE= 254.80 ELECTRON DENSITY= 0.20E 24 

A=0.40 R- O.IOE 00 TRATIO=1.00 Xl=17.0 21= 1.0 Z2=17.0 



DELTA 


STARK 


STARK+ 


OMEGA 


PROFILE 


OOPPLER 


0.0 


. 4 1 7E 


02 


0.U8E 2 


0.735E-02 


0. 138E 


02 


0.112E 02 


0. 147E-0 1 


0.49 8E 


01 


0.967E 01 


0.22 IE-0 1 


0.273E 


01 


0.762E 01 


0.29 4E-0 1 


0.204E 


01 


0.559E 01 


0.368E-01 


0.175E 


01 


0.396E 1 


0.44 1E-01 


O. 163E 


01 


0.283E 01 


0.515E-01 


0. 157E 


01 


0.214E 01 


0.58 8E-01 


0. 153E 


01 


0.1 76E 1 


0.662E-0 I 


0.1 47E 


01 


0.15 4E 1 


0.735E-01 


0. 14IE 


1 


0.141E 01 


0.882E-01 


0. 123E 


01 


0-122E 01 


0.I0 3E 


0. 104E 


01 


0.104E 01 


0.I18E 


0.849E 


00 


0.867E 00 


0.132E 00 


0.68 0E 


00 


0.705E 00 


. 1 4 7E 


0.539E 


00 


0.564E 00 


0.221E 00 


0. 169E 


00 


0.1 79E 


0.294E 00 


0.632E- 


-01 


0.657E-01 


0.368E 00 


0.291E- 


-01 


0.29BE-01 


0.441E 00 


0. 158E- 


-01 


0.161E-01 


0.58 8E 


0.655E- 


-02 


0.662E-02 


0.735E 00 


0.364E- 


-02 


0.366E-02 



LYMAN ALPHA PROFILE FOR HYDROGENIC ARGON 
ELECTRON TEMPERATURE- 254.80 ELECTRON DENSITY* 0. 20E 24 

A=0.40 R= O.IOE 13 TRATIO=1.00 XI=17.0 Zl= 1.0 Z2=17.0 



DELTA 


STARK 


STARK+ 


OMEGA 


PROFILE 


DOPPLER 


0.0 


0.418E 


02 


0.H9E 02 


0.735E-02 


0. 139E 


02 


0. 113E 02 


0. 147E-01 


0.505E 


01 


0.974E 01 


0.22 1E-01 


0.287E 


01 


0.769E 1 


0.294E-01 


0.213E 


01 


0.567E 1 


0.368E-0 1 


0. 185E 


01 


0.404E 1 


0.441E-0 1 


0. 172E 


01 


0.290E 01 


0.515E-0 1 


0.165E 


01 


0.220E 01 


0.588E-01 


0. 158E 


01 


0.1 80E 01 


0.662E-01 


0.151E 


01 


0.158E 01 


0.735E-01 


0. 142E 


01 


0.143E 01 


0.882E-01 


0. 122E 


01 


0.1 22E 1 


0.10 3E 


O. 101E 


01 


0.I02E 01 


. 1 1 8£ 


0. 816E 


00 


0.839E 00 


0.132E 00 


0.648E 


00 


0.675E 00 


0.14 7E 


0.510E 


00 


0.536E 00 


0.221E 00 


0. 156E 


00 


0.1 64E 


0.294E 00 


C.558E- 


-01 


0.582E-01 


0.368E 00 


0.248E- 


-01 


0.255E-01 


0.441E 00 


0.133E- 


-01 


0. 135E-01 


0.588E 00 


0.552E- 


-02 


0.558E-02 


0.735E 00 


0.319E- 


-02 


0.321E-02 



242 



LYMAN BETA PROFILE FOR HYDROGENIC ARGON 
ELECTRON TEMPERATUR£= 809,10 ELECTRON OENSITY= 0. 10E 24 



A=0.20 Rat 0.0 



TRATI0=0.2S XI=17.0 Z 1 = 1.0 Z2=17.0 



DELTA 


STARK 




STARK+ 


OMEGA 


PROFILE 


DOPPLER 


0.0 


0.262E 


00 


0. 1 10E 01 


0.735E-02 


0.285E 


00 


0.110E 01 


0.14 7E-0 1 


0.346E 


00 


0,1 I IE 01 


0.22 IE-01 


0.435E 


00 


0.1 1 IE 01 


0.29 4E-O1 


0.547E 


00 


0.112E 01 


0*36 8E-0 t 


0.6 7 7E 


00 


0. 1 14E 1 


0.441E-01 


0.818E 


00 


0.115E 01 


0.515E-0 1 


0.965E 


00 


0-1 17E 1 


0.588E-01 


0. 11 IE 


01 


0.119E 01 


0.662E-01 


0.125E 


01 


0.121E 01 


0.735E-01 


0, 137E 


01 


0.123E 1 


0,882E-01 


0. 158E 


01 


0,1 27E 01 


G.I03E 00 


0. 170E 


01 


0,131E 01 


0.U8E 


0,1 75E 


01 


0.134E 01 


0.132E 00 


0,174E 


01 


0.135E 01 


0.14 7E 


0, 168E 


01 


0.136E 01 


0.221E 00 


0, 116E 


01 


0.119E 01 


0.29 4E 


0-743E 


00 


0,868E 00 


0.368E 00 


0.484E 


00 


0,582E 00 


0.441E 00 


0.323E 


00 


0.387E 00 


0.588E 00 


0. 158E 


00 


0.184E 00 


0.735E 00 


0.874E- 


-01 


0.990E-0 1 



LYMAN BETA PROFILE FOR HYDROGENIC ARGON 
ELECTRON TEMPERATURE^ 809*10 ELECTRON DENSITY= O.IOE 24 



A=0.20 R= 0.0 



TRATIO=1.00 XI=17.0 Z 1= 1.0 Z2=17.0 



DELTA 
OMEGA 

0.0 

0.735E-02 
0.147E-0 1 
0.221E-0 1 
0.294E-01 
0.368E-01 
0.441E-01 
0.51 5E-01 
0.588E-01 
0.662E-01 
0.73 5E-0 1 
0.882E-01 
. I 3E 
0.11 8E 
0.132E 00 
. I 4 7E 
0.221E 00 
0.294E 00 
0.36 8E 
0.441E 00 
0.58 8E 
0.73 5E 



STARK 
PROFILE 



STARK+ 
DOPPLER 



0.312E 
0.34 0E 
0.417E 
0.S28E 
0.666E 
0.824E 
0.995E 
0, 1 17E 
O. 134E 
0,1 49E 
0, 163E 
0.183E 
0. 193E 
0. 194E 
0. 188E 
O. 178E 
. 1 1 3E 
0.691E 
0.430E 
0.276E 
0. 126E 



00 
00 
00 
00 
00 
00 
00 
01 
01 
01 
01 
01 
01 
01 
01 
01 
01 
00 
00 
00 
00 



0.834E 
0.844E 
0.871E 
0.915E 
0.974E 
0.104E 
0.1 12E 
0.121E 
0.130E 
0.1 39E 
0.147E 
0. 161E 
0,171E 
0.1 76E 
0.1 76E 
0.1 72E 
0.121E 
0.750E 
0.466E 
0.298E 
0.135E 



00 
00 
00 
00 
00 
01 
01 
01 
01 
01 
01 
01 
01 
01 
01 
01 
01 
00 
00 
00 
00 



0.671E-01 



0.711E-01 



243 



LYMAN BETA PROFILE FOR HYOROGENIC ARGON 
ELECTRON TEMPERATURE^ 809.10 ELECTRON DENSITY^ 0. I OE 24 



A=0.2 R= 0.0 



TRATIO=4.00 XI=17.0 Zl = 1.0 Z2=17.0 



DELTA 


STARK 


STARK+ 


OMEGA 


PROFILE 


DOPPLER 


0.0 


0.499E 


00 


0.82 9E 


0.735E-02 


0.552E 


00 


0.859E 00 


0. 147E-0 1 


0.692E 


00 


0.946E 00 


0.22 1E-0 1 


0.892E 


00 


0.108E 01 


0.29 4E-0 1 


0.113E 


01 


0.125E 01 


0.368E-0 1 


0. 140E 


01 


0. 145E I 


0.44 1E-Q 1 


0. 167E 


01 


0.165E 01 


0.51 5E-0 1 


0. 192E 


01 


0.185E 01 


0.588E-01 


0.214E 


01 


0.203E 01 


0.662E-Q1 


0.232E 


01 


0.219E 01 


0.735E-01 


0.245E 


01 


0.231E 01 


0.882E-01 


0.256E 


01 


0.244E 1 


0.103E 00 


0.250E 


01 


0.24 3E 1 


0.118E 


0.233E 


01 


0.231E 01 


0.132E 00 


0.211E 


01 


0.212E 01 


0.147E 00 


0.187E 


01 


0.191E 01 


0.221E 00 


0.954E 


00 


0.990E 00 


0.294E 00 


0.499E 


00 


0.51 7E 


0.368c 00 


0.270E 


00 


0.2 79E 


0.441E 00 


0. 154E 


00 


0.159E 00 


0.58 8E 


0.621E- 


-0 1 


0.638E-01 


0.73 5E 


0.31SE- 


-01 


0.322E-01 



LYMAN SETA PROFILE FOR HYOROGENIC ARGON 
ELECTRON TEMPERATURE^ 809.10 ELECTRON DENSITY= 0.10E 24 

A=0.20 R= 0.10E 00 TRATIO=1.00 XI=17.0 Z 1= 1.0 Z2=17.0 



DELTA 

OMEGA 



0.0 

0.73 

0.14 

0.22 

0.29 

0.36 

0.44 

0.51 

0.58 

0.66 

0.73 

0.88 

0.10 

0.11 

0. 13 

0.14 

0.22 

0.29 

0.36 

0.44 

0.58 

0.73 



5E-0 2 
7E-0 1 
1E-0 1 
4E-0 1 
8E-0 1 
1E-0 1 
5E-01 
8E-0 1 
2E-0 1 
5E-0 1 
2E-0 1 
3£ 



8E 
2E 
7E 
IE 
4E 
8E 
IE 
8E 
5E 



00 
00 
00 
00 
00 
00 
00 
00 
00 



STARK 
PROFILE 

0.232E 00 

0.251E 00 

0.303E 00 

0.378E 00 

0.473E 00 

0.S85E 00 

0.707E 00 

0.836E 00 

0.965E 00 

0. 109E 01 

0.121E 01 

0.141E 01 

0.1 55E 01 

0.162E 01 

0.164E 01 

0.162E 01 

0.122E 01 

. 8 1 1 E 

0.538E 00 

0.362E 00 

0.1 74E 
0.915E-01 



STARK+ 
DOPPLER 

0.61 7E 
0.624E 00 
0.646E 00 
0.682E 00 
0.729E 00 
0.788E 00 
0.855E 00 
0.928E 00 
O.IOOE 01 
0.1 08E 1 
0.116E 01 
0.130E 1 
0.142E 01 
0. 151E 01 
0.155E 01 
0.156E 01 
0.128E 01 
0.875E 00 
0.582E 00 
0.391E 00 
0.188E 00 
0.980E-01 



244 



LYMAN BETA PROFILE FOR HYDROGEN IC ARGON 
ELECTRON TEMPERATURE^ 809-10 ELECTRON OENSITY= O.IOE 24 

A=0.20 R= O.IOE 13 TRATIO=1.00 Xl=17.0 Zl= UO Z2=17.0 



DELTA 


STARK 




STARK+ 


OMEGA 


PROFILE 


DOPPLER 


0.0 


0.206E 


00 


0.549E 00 


Q.735E-02 


0.223E 


00 


0.555E 00 


0.147E-01 


0.267E 


00 


0.575E 00 


0.221E-0 1 


0.332E 


00 


0.608E 00 


0.294E-0 1 


0.4 15E 


00 


0.651E 00 


0.368E-0 1 


0.S11E 


00 


0.705E 00 


0.44 1E-01 


0.6 I 9E 


00 


0.767E 00 


0.515E-01 


0. 732E 


00 


0.835E 00 


0.588E-0 1 


0.848E 


00 


0.906E 00 


0.t>62E-01 


0.961E 


00 


0.980E 00 


0.735E-01 


0. 107E 


01 


0.1 05E 1 


0.882E-01 


0.1 26E 


01 


0.1 19E 1 


0.103E 00 


0. 140E 


01 


0.131E 01 


0.1 1 8E 


0. 149E 


01 


0.140E 01 


0.13 2E 


0. 153E 


01 


0.146E 01 


0.14 7E 


0. 153E 


01 


0.149E 01 


0.221E 00 


0. 122E 


01 


0.128E 01 


0.294E 00 


0.848E 


00 


0.916E 00 


0.368E 00 


0.578E 


00 


0.628E 00 


0.44 IE 00 


0.398E 


00 


0.432E 00 


0.58 8E 


0. 19 7E 


00 


0.214E 00 


0.735E 00 


0. 105E 


00 


0.113E 



LYMAN BETA PROFILE FOR HYDROGENIC ARGON 
ELECTRON TEMPERATURE^ 202.30 ELECTRON DENSITY= O.IOE 24 



A=0.40 R= 0.0 



TRATIO=0.25 Xl=17.0 21= 1.0 Z2=17.0 



DELTA 


STARK 




STARK+ 


OMEGA 


PROFILE 


DOPPLER 


0.0 


0.487E 


00 


0.102E 01 


0.735E-02 


0.518E 


00 


0.103E 1 


0. 147E-0 1 


0.602E 


00 


0.106E 01 


0.221E-01 


0.724E 


00 


0. 1 10E 01 


0.294E-0 I 


0.871E 


00 


0. 115E 01 


0.368E-0I 


0.1 03E 


01 


0.12LE 01 


0.44 1E-0 1 


0. 120E 


1 


0.128E 01 


0.51 5E-01 


0. 136E 


01 


0.136E 01 


0.S8 8E-01 


0.151E 


01 


0.143E 01 


0.662E-0 1 


0. 164E 


01 


0. 151E 01 


0.73SE-01 


0. 175E 


01 


0.1 57E 1 


0.882E-01 


0. 189E 


01 


0.168E 01 


0.10 3E 


0. 193E 


01 


0.1 74E 1 


. 1 1 8E 


0. 189E 


01 


0.176E 01 


0.132E 00 


0. 180E 


01 


0.I73E 1 


. 1 4 7E 


0. 168E 


01 


0.167E 01 


0.221E 00 


0. 106E 


01 


0. 114E 01 


0.294E 00 


0.647E 


00 


0.706E 00 


0.368E 00 


0.4 07E 


00 


0.442E 00 


0.441E 00 


0.266E 


00 


0.287E 00 


0.58 8E 


0. 128E 


00 


0.1 37E 


0.73 5E 


0.707E- 


-01 


0.752E-01 



245 



LYMAN BETA PROFILE FOR HYOROGENIC ARGON 
ELECTRON TEMP£RATURE= 202.30 ELECTRON DENSITY^ O.IOE 24 



A=0.40 R~ 0.0 



TRATIO=1.00 Xl=17.0 Zl = 1.0 Z2=17.0 



DEL.TA 


i 


STARK 


STARK+ 


OMEG/ 


i 


PROFILE 


DOPPLER 


0.0 




0.732E 


00 


0.106E 01 


0.735E- 


•02 


0. 783E 


00 


0.109E 01 


0.14 7E- 


•0 1 


0.922E 


00 


0. 1 17E 1 


0.22 1E- 


01 


0.1 12E 


01 


0.129E 01 


0.294E- 


•01 


0. 1 35E 


01 


0.1 45E 1 


0.368E- 


■01 


0. 159E 


01 


0.162E 01 


0.44 ie- 


•0 1 


0. 183E 


01 


0.180E 01 


0.51 5E- 


■01 


0.203E 


01 


0.196E 01 


0.58 8E- 


•01 


0.221E 


01 


0.21 IE 01 


0.662E- 


■01 


0.234E 


01 


0.222E 01 


0.735E- 


•01 


0.242E 


01 


0.231E 01 


0.882E- 


■01 


0.245E 


01 


0.237E 01 


0. 10 3E 


00 


0.235E 


1 


0.232E 01 


0. 1 1 8E 


00 


0.217E 


01 


0.21 8E 1 


0. 132E 


00 


0. 196E 


01 


0.1 99E 1 


- 1 4 7E 


00 


0.1 74E 


01 


0.179E 01 


0.22 IE 


00 


0.909E 


00 


0.947E 00 


0.294E 


00 


0.485E 


00 


0.506E 00 


0.368E 


00 


0.273E 


00 


0.284E 00 


0.44 IE 


00 


0. 165E 


00 


0.171E 00 


0.588E 


00 


0.721E- 


-01 


0.744E-01 


0.735E 


00 


0.387E- 


-01 


0.398E-01 



LYMAN BETA PROFILE FOR HYOROGENIC ARGON 
ELECTRON TEMPERATURE^ 202.30 ELECTRON DENSITY= 0.10E 24 



A=0.40 


R= O.C 


1 


TRAT10=4.0( 


) xi=i; 


'.0 Z\~ 1.0 




DELTA 


STARK 


STARK4- 




OMEGA 


PROFILE 


OOPPLER 




0.0 




0. 155E 


01 


0.181E 01 




0.735E- 


■02 


0.168E 


01 


0.191E 01 




0. 14 7E- 


-0 1 


0.202E 


01 


0.21 6E 1 




0.22 IE- 


01 


0.245E 


01 


0.250E 01 




Q.29 4E- 


■0 1 


0.288E 


01 


0.288E 01 




0.368E- 


■01 


0.325E 


01 


0.320E 01 




0.44 1E- 


■01 


0.351E 


01 


0.344E 01 




0.51 5E- 


■01 


0.364E 


01 


0.358E 01 




0.58 8E- 


-01 


0.364E 


01 


0.359E 1 




0.662E- 


■01 


0.353E 


01 


0.352E 01 




0.735E- 


-01 


0.335E 


01 


0.336E 01 




0.882E- 


•01 


0.288E 


01 


0.292E 01 




0. 103E 


00 


0.237E 


01 


0.243E 01 




0. U8E 


00 


0. 193E 


01 


0.199E 01 




0.13 2E 


00 


0. 157E 


01 


0.162E 01 




. 1 4 7E 


00 


0.128E 


01 


0.1 32E 1 




0.22 IE 


00 


0.486E 


00 


0.500E 00 




0.294E 


00 


,21 8E 


00 


0.223E 00 




0.368E 


00 


„ 11 9E 


00 


0.122E 00 




0.441E 


00 


0„754E- 


-01 


0.769E-01 




0.58 8E 


00 


0..389E- 


-01 


0.396E-0 1 




0.73 5c 


00 


0„ 241E- 


-01 


0.246E-01 



246 



LYMAN BETA PROFILE FOR HYDROGEN IC ARGON 
ELECTRON TEMPER ATURE= 202.30 ELECTRON DENSITY^ 0.10E 24 

A=0.40 R= 0.10E 00 TRATIO=1.00 XI=17.0 Zl = 1*0 Z2=17.0 



DELTA 


STARK 


STARK+ 


OMEGA 


PROFILE 


OOPPLER 


0.0 


0.844E 


00 


0.122E 01 


0.735E-02 


0.906E 


00 


0.125E 01 


0. 147E-0 1 


0. 107E 


01 


0.1 34E 1 


0.22 1E-01 


0.130E 


01 


0.147E 01 


0-294E-0 1 


0. 156E 


01 


0.1 64E 1 


0.36 8E-01 


0. 182E 


01 


0.182E 01 


0.44 IE-01 


0.207E 


0! 


0.200E 01 


0.51 5E- 1 


0.227E 


01 


0.216E 1 


0.588E-01 


0.242E 


01 


0.229E 01 


0.662E-0 1 


0.253E 


01 


0.239E 01 


0.735E-01 


0.257E 


01 


0.245E 01 


0.88 2E-01 


0.254E 


01 


0.247E 01 


0.103E 00 


0.239E 


01 


0.236E 01 


0. 1 I 8E 


0.2I7E 


01 


0.21 8E 1 


0.132E 00 


0. 193E 


01 


0.197E 01 


0.14 7E 


0.1 70E 


01 


0.1 75E 1 


0.221E 00 


0.853E 


00 


0.889E 00 


0.294E 00 


0.443E 


00 


0.461E 00 


0.368E 00 


0.245E 


00 


0.255E 00 


0.441E 00 


0. 146E 


00 


0. 151E 00 


0.588E 00 


0.62 7E- 


-01 


0.645E-0 1 


0.735E 00 


0.335E- 


-01 


0.344E-0 1 



LYMAN BETA PROFILE FOR HYDROGEN IC ARGON 
ELECTRON TEMPERATURE^ 202.30 ELECTRON DENSITY= 0.10E 24 

A=0.40 R- 0.10E 13 TRATIO=1.00 XI=17.0 Zl = 1.0 Z2=17.0 



DELT/1 


L 


STARK 


STARK* 


OMEGA 


PROFILE 


DOPPLER 


0.0 




0.925E 


00 


0.1 33E 1 


0.73 5E- 


■02 


0.994E 


00 


0.136E 1 


0.147E- 


■01 


0.1 I 8E 


01 


0.145E 01 


0.22 IE- 


01 


0. 143E 


01 


0.1 59E 1 


0.29 4E- 


■01 


0. 171E 


01 


0.1 77E 1 


0.368E- 


01 


0.1 98E 


01 


0.1 95E 1 


0.44 1£- 


■0 1 


0.222E 


01 


0.213E 01 


0.51 5E- 


■0 1 


0.242E 


01 


0.228E 1 


0.58 8E- 


01 


0.256E 


01 


0.241E 01 


0.662E- 


■01 


0.264E 


Oi 


0.249E 01 


0.73 5E- 


•01 


0.266E 


01 


0.254E 01 


0.88 2E- 


-01 


0.259E 


01 


0.252E 01 


0.103E 


00 


0.240E 


01 


0.238E 01 


0.1 18E 


00 


0.21 6E 


Oi 


0.218E 1 


0. 132E 


00 


0. 191E 


01 


0.195E 1 


. 1 4 7E 


oo 


0. 167E 


01 


0.172E 01 


0.221E 


00 


. 8 1 7E 


00 


0.852E 00 


0.294E 


00 


, 4 1 7E 


00 


0.434E 00 


0.36 8E 


00 


0.228E 


00 


0.237E 00 


0.441E 


00 


0, 134E 


00 


0.139E 00 


0.58 8E 


00 


Q.572E- 


-01 


0.588E-0 1 


0.735E 


00 


0.3 07E- 


-01 


0.314E-01 



LIST OF REFERENCES 



1. H. R. Griem, Plasma Spectroscopy (McGraw-Hill Book Company, New 
York, 1964) . 

2. M. Baranger, Atomic and Molecular Processes , D. R. Bates, Ed. 
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3. J. Cooper, Plasma Spectroscopy, Rep. Progr. Phys . _29, 35 (1966). 

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Electron Atoms , (Springer Verlag, Berlin, 1957). 

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17. E. W. Smith, Dissertation (University of Florida, 1966). 

247 



248 

18. T. W. Hussey, Dissertation (University of Florida, 1974). 

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20. J. E. Whalen, Dissertation (University of Florida, 1972). 

21. James W. Dufty, Phys. Rev. 187, 305 (1969). 

22. E. W. Smith, Phys. Rev. 166, 102 (1968). 

23. John T. O'Brien, Dissertation (University of Florida, 1970). 

24. C. F. Hooper, Jr., Phys. Rev. 165, 215 (1968). 

25. C. F. Hooper, Jr., Phys. Rev. 149, 77 (1966). 

26. Phillip M. Morse and Herman Feshbach, Methods of Theoretical Physics 
(McGraw-Hill Book Company, New York, 1953). 

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28. C. F. Hooper, Jr., Phys. Rev. 169 , 193 (1968). 

29. J. Auerbach, Lawrence Livermore Laboratory internal memorandum 
LPIG-77-34, February 7, 1977 (unpublished). 

30. Bernard Mozer and Michel Baranger, Phys. Rev. 118 , 626 (1960). 

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32. Hugh E. DeWitt, Low-Luminosity Stars (Gordon and Breach, New 
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34. H. R. Griem, preprint (March, 1977). 

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249 

41. A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton 
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BIOGRAPHICAL SKETCH 

Richard Joseph Tighe was born on July 6, 1946, in Columbia, 
South Carolina. He graduated from A.C. Flora High School in June, 
1964. In June, 1969, he received the Bachelor of Science degree 
with a major in Physics from the University of South Carolina. He 
spent the summer of 1969 working in the Stable Isotopes Separation 
Division of Union Carbide at Oak Ridge, Tennessee. In September, 
1969, he enrolled in the Graduate School of the University of Florida. 
From that time until the present he has worked toward the degree of 
Doctor of Philosophy. 

Richard Joseph Tighe is married to Janette Cornish Gervin. 
He is a member of the American Physical Society. 



250 



I certify that I have read this study and that in my opinion it 
conforms to acceptable standards of scholarly presentation and is 
fully adequate, in scope and quality, as a dissertation for the degree 
of Doctor of Philosophy. 




C. F; Hooper, //Jr. , Chairman 
Professor of physics 



I certify that I have read this study and that in my opinion it 
conforms to acceptable standards of scholarly presentation and is 
fully adequate, in scope and quality, as a dissertation for the degree 
of Doctor of Philosophy. 




E. D. Adams 
Professor of Physics 

I certify that I have read this study and that in my opinion it 
conforms to acceptable standards of scholarly presentation and is 
fully adequate, in scope and quality, as a dissertation for the degree 
of Doctor of Philosophy. 




P. W. Vickers 

Associate Professor of Computer and 
Information Sciences 

This dissertation was submitted to the Graduate Faculty of the 
Department of Physics in the College of Arts and Sciences and to the 
Graduate Council, and was accepted as partial fulfillment of the 
requirements for the degree of Doctor of Philosophy. 

June 1977 



Dean, Graduate School